Mathematical Researches
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کران بالایی برای مرتبهی یک خانوادهی ضربدری -tتقریباً اشتراکی
An upper bound for the size of a cross t-almost ntersecting family
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مقاله مستقل
Original Manuscript
فرض کنید <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > خانوادهای از زیرمجموعههای <em><span dir="LTR">k</span></em> عضوی از یک مجموعه <em><span dir="LTR">n</span></em> عضوی <em><span dir="LTR">X</span></em> باشد. به <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > اشتراکی گویند هرگاه برای هر دو عضو <em><span dir="LTR">A</span></em> و <em><span dir="LTR">B</span></em> متعلق به <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > داشته باشیم <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image002.png" > . قضیهی معروف اردوش-کو-رادو بیان میکند اندازهی یک خانواده اشتراکی از زیرمجموعههای <em><span dir="LTR">k</span></em> عضوی از یک مجموعه <em><span dir="LTR">n</span></em> عضوی حداکثر <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image003.png" > است و تساوی زمانی برقرار است اگر و تنها اگر <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image004.png" > عضوی مانند <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image005.png" > وجود داشته باشد که برای هر عضو در <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > مانند <em><span dir="LTR">A</span></em> <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image006.png" > داشته باشیم <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image007.png" > . فرض کنید <em><span dir="LTR">k</span></em> و <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image008.png" > دو عدد صحیح مثبت باشند که <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image009.png" > . فرض کنید <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > خانوادهای از زیرمجموعههای <em><span dir="LTR">k</span></em> عضوی از مجموعه <em><span dir="LTR">n</span></em> عضوی <em><span dir="LTR">X</span></em> و <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image010.png" > خانوادهای از زیرمجموعههای <em><span dir="LTR">ℓ</span></em> عضوی از مجموعه <em><span dir="LTR">X</span></em> باشد به دو خانواده <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > و <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image010.png" > دو خانواده ضربدری <span dir="LTR">–</span><em><span dir="LTR">t</span></em>تقریباً اشتراکی گویند اگر هر عضو <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > با حداکثر <em><span dir="LTR">t</span></em> عضو از خانواده <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image010.png" > اشتراک نداشته باشد و همینطور <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image011.png" > با حداکثر <em><span dir="LTR">t</span></em> عضو از خانواده <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > اشتراک نداشته باشد در این مقاله به عنوان تعمیمی از قضیهی اردوش-کو-رادو نشان میدهیم اگر <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > و <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image010.png" > دو خانواده ضربدری <span dir="LTR">–<em>t</em></span> تقریباً اشتراکی باشند و <em><span dir="LTR">n</span></em> به اندازه کافی بزرگ باشد، آنگاه<br>
<img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image012.png" ><br>
و تساوی زمانی رخ می دهد اگر و تنها اگر عضوی مانند <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image005.png" > وجود داشته باشد که برای هر عضو <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image013.png" > متعل به <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" > و هر عضو <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image014.png" > متعلق به <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image010.png" > داشته باشیم، <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image007.png" > و <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image015.png" > .
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span style="font-family:NimbusRomNo9L-Regu">Introduction</span></span></b></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Let </span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">n</span></span></i> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">and <i>k</i> be two positive integers such that </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>≥</m:r><m:r>k</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Let </span></span><m:omath><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r></span></span></i></m:e></m:d><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>{1,...,</m:r><m:r>n</m:r><m:r>}</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be an </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">-elemen set and let symbol </span></span><m:omath><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="1"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:mpr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:e></m:mr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r></span></span></i></m:e></m:mr></m:m></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r> </m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">denote the family of all <i>k</i>-element subsets (or <i>k</i>-sets) of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. A family </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> of <i>k</i>-sets of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> is said to be intersecting if for every two members <i>A, B</i> in </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt">,</span></span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> we have </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>A</m:r><m:r>∩</m:r><m:r>B</m:r><m:r>≠∅</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. For a fixed element </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r><m:r>∈[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, if all members of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> contain <i>i</i>, then it is clear that </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> is an intersecting family, which is called star. For each</span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr> </m:r><m:r>i</m:r><m:r>∈[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, the family</span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r> </m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r></span></span></i></m:sub></m:ssub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>={</m:r><m:r>A</m:r><m:r>:</m:r></span></span></i><m:d><m:dpr><m:begchr m:val="|"><m:endchr m:val="|"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>A</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>=</m:r><m:r>k</m:r><m:r>, </m:r><m:r>A</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>⊆</m:r></span></span></i><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>, </m:r><m:r>i</m:r><m:r>∈</m:r><m:r>A</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">} is a maximal star.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> The well-known </span></span><em><span style="font-size:10.0pt"><span style="background:white"><span style="color:#5f6368">Erdős</span></span></span></em> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">–Ko–Rado theorem is one of important results in extremal combinatorics. It has many interesting proofs and extensions.</span></span> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">The </span></span><em><span style="font-size:10.0pt"><span style="background:white"><span style="color:#5f6368">Erdős</span></span></span></em><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">-Ko-Rado theorem states that every intersecting family of </span></span><m:omath><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="1"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:mpr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:e></m:mr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r></span></span></i></m:e></m:mr></m:m></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> has cardinality at most </span></span><m:omath><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="1"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:mpr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e></m:mr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e></m:mr></m:m></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> provided that </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>≥2</m:r><m:r>k</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">; moreover, if </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>>2</m:r><m:r>k</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, then the only intersecting families of this cardinality are isomorphic to </span></span><m:omath><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. </span></span><span style="font-size:10.0pt"><span style="font-family:"Helvetica",sans-serif"></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Two families </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>⊆</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="1"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r></span></span></i></m:e></m:d></m:e></m:mr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r></span></span></i></m:e></m:mr></m:m></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B⊆</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="1"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r></span></span></i></m:e></m:d></m:e></m:mr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></span></span></span></i></m:e></m:mr></m:m></m:e></m:d></m:omath> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt">are called </span></span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">cross intersecting if for any </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>A</m:r><m:r>∈</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>B</m:r><m:r>∈</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B</m:r></span></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt">, </span></span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">we have </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>A</m:r><m:r>∩</m:r><m:r>B</m:r><m:r>≠∅</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Strengthening the Erdős–Ko–Rado theorem, in 1986 Pyber showed an upper bound for </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>|</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>||</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>|</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> as follows.<span style="letter-spacing:-.3pt"></span></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Theorem A</span></span></b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Let <i>k</i>, </span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">ℓ</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, <i>n</i> be positive integers such that </span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">k</span></span></i><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>≥</m:r></span></span></i></m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">ℓ</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Assume that </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>⊆</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and</span></span> <m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B⊆</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">are cross intersecting.</span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span>
<ul>
<li style="margin-left:8px"><span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">If </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> <i>k></i></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">ℓ</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>≥2</m:r><m:r>k</m:r><m:r>+</m:r></span></span></i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr>l-</m:r><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>2</m:r></span></span></m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, </span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">then<i> </i></span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>|</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>||B|≤</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>.</m:r></span></span></i></m:omath><span dir="RTL" lang="AR-SA" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></li>
<li style="margin-left:8px"><span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">If </span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">k=</span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">ℓ</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>≥2</m:r><m:r>k</m:r></span></span></i></m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, </span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">then<i> </i></span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>|</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>||B|≤</m:r></span></span></i><m:ssup><m:ssuppr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssuppr><m:e><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:e><m:sup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>2</m:r></span></span></i></m:sup></m:ssup><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>.</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></li>
</ul>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"> </span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> In 1989 Matsumoto and Tokushige slightly improved Pyber's result as follows.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Theorem B</span></span></b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Let <i>k, </i></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">ℓ, n</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be positive integers such that<i> </i></span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>≥2 </m:r></span></span></i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>max</m:r><m:r><i>{</i></m:r><m:r><i>k</i></m:r></span></span></m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">,</span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> ℓ}</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Assume that </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>⊆</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and</span></span> <m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B⊆</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">are cross intersecting. Then </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>|</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>||B|≤</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>.</m:r></span></span></i></m:omath> </span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> We say that a family </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> is <i>t</i>-almost intersecting if for every set </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>A</m:r><m:r>∈</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> there are at most <i>t</i> elements of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span></span></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">disjoint from </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>A</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. In 2012 </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Gerbner, Lemons, Palmer, Patkós, and Szécsi</span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> proved an interesting generalization of the Erdős–Ko–Rado theorem for <i>t</i>-almost intersecting families. </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Theorem C</span></span></b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Let <i>k</i></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, n. t</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be positive integers and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>⊆</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r></span></span></i></m:e></m:d></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r></span></span></i></m:e></m:eqarr></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r> </m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">is a <i>t</i>-almost intersecting family.</span></span></span></span></span>
<ul>
<li style="margin-left:8px"><span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">If </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> <i>n=n</i>(<i>k,t</i>) is sufficiently large, then </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>|</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>|≤</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">with equality if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>=</m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"> for some </span></span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>∈[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:omath><span dir="RTL" lang="AR-SA" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"></span></span></span></span></span></li>
<li style="margin-left:8px"><span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">If </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r> </m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r><m:r>≥3</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>≥2</m:r><m:r>k</m:r><m:r>+2</m:r></span></span></i></m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">, and </span></span></i><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>t</m:r><m:r>=1</m:r></span></span></i></m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">,</span></span></i><i> </i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">then </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>|</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>|≤</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">with equality if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>=</m:r></span></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"> for some </span></span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>∈[</m:r><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">].</span></span></span></span></span></li>
</ul>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="text-autospace:none"><span style="font-family:Calibri,sans-serif"> <b><span style="font-size:10.0pt">Main Results</span></b></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> We say two families </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>⊆</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="1"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r></span></span></i></m:e></m:d></m:e></m:mr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r></span></span></i></m:e></m:mr></m:m></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B⊆</m:r></span></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:m><m:mpr><m:mcs><m:mc><m:mcpr><m:count m:val="1"><m:mcjc m:val="center"></m:mcjc></m:count></m:mcpr></m:mc></m:mcs><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r></span></span></i></m:e></m:d></m:e></m:mr><m:mr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></span></span></span></i></m:e></m:mr></m:m></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> are cross <i>t</i>-almost intersecting if any </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>A</m:r><m:r>∈</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> is disjoint from at most <i>t</i> elements of </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and any </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>B</m:r><m:r>∈</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> is disjoint from at most <i>t</i> elements of</span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr> </m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. As our main result we simultaneously extend the previous results for sufficiently large <i>n</i>.</span></span></span></span></span><br>
<span style="font-size:11pt"><span style="line-height:21.0pt"><span style="font-family:Calibri,sans-serif"><b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">Theorem 1</span></span></b><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Let <i>k, </i></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">ℓ, n</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> be positive integers such that</span></span><i> </i><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">n=n</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> (<i>k,</i></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> ℓ</span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> ,t</span></span></i><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">) is sufficiently large. Assume that </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></span></i><span dir="RTL" style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="p"></m:sty></m:scr></m:rpr> </m:r></span></span></span><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>⊆</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> and</span></span> <m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B⊆</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>[</m:r><m:r>n</m:r><m:r>]</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath> <span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">are </span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">cross <i>t</i>-almost intersecting</span></span><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">. Then </span></span><m:omath><m:d><m:dpr><m:begchr m:val="|"><m:endchr m:val="|"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></span></span></i></m:e></m:d><m:d><m:dpr><m:begchr m:val="|"><m:endchr m:val="|"><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:endchr></m:begchr></m:dpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B</m:r></span></span></i></m:e></m:d><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>≤</m:r></span></span></i><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d><m:d><m:dpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:dpr><m:e><m:eqarr><m:eqarrpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:eqarrpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></span></span></i></m:e><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>-</m:r><m:r>1</m:r></span></span></i></m:e></m:eqarr></m:e></m:d></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"> with equality if and only if </span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="letter-spacing:-.3pt"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>=</m:r></span></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B</m:r><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>=</m:r></span></span></i><m:ssub><m:ssubpr><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><span style="font-style:italic"><m:ctrlpr></m:ctrlpr></span></span></span></m:ssubpr><m:e><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>S</m:r></span></span></i></m:e><m:sub><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r></span></span></i></m:sub></m:ssub></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif"><span style="letter-spacing:-.3pt"> for some </span></span></span><m:omath><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>i</m:r></span></span></i><i><span style="font-size:10.0pt"><span style="font-family:"Cambria Math",serif"><m:r>∈[</m:r><m:r>n</m:r></span></span></i></m:omath><span style="font-size:10.0pt"><span style="font-family:"Times New Roman",serif">].</span></span></span></span></span><br>
قضیه اردوش-کو-رادو, خانواده اشتراکی, خانواده ضربدری اشتراکی
Erdős-Ko-Rado theorem, Intersecting family. Cross intersecting family
205
214
http://mmr.khu.ac.ir/browse.php?a_code=A-10-698-1&slc_lang=fa&sid=1
Ali
Taherkhani
علی
طاهرخانی
ali.taherkhani@gmail.com
10031947532846005773
10031947532846005773
Yes
Institute for Advanced Studies in Basic Sciences (IASBS)
دانشکده ریاضی، دانشگاه تحصیلات تکمیلی علوم پایه زنجان