Mathematical Researches

fa کران بالایی برای مرتبه‌ی یک خانواده‌ی ضرب‌دری -tتقریباً اشتراکی An upper bound for the size of a cross t-almost ntersecting family ریاضی Mat مقاله مستقل Original Manuscript فرض کنید <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" >  خانواده‌ای از زیرمجموعه‌های k عضوی از یک مجموعه n عضوی X باشد. به <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" >  اشتراکی گویند هرگاه برای هر دو عضو A و B 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 k عضوی از یک مجموعه n 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 A <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 k 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 k عضوی از مجموعه n عضوی X و <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image010.png" >  خانواده‌ای از زیرمجموعه‌های ℓ عضوی از مجموعه X 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 –tتقریباً اشتراکی گویند اگر هر عضو <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image001.png" >  با حداکثر t 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 t 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 –t تقریباً اشتراکی باشند و n به اندازه کافی بزرگ باشد، آنگاه <img alt="" chromakey="white" src="file:///C:Usersuser1AppDataLocalTempmsohtmlclip1�1clip_image012.png" > 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" > . Introduction Let n and k be two positive integers such that <m:omath><m:r>n</m:r><m:r>≥</m:r><m:r>k</m:r></m:omath>. Let <m:omath><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>n</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="roman"><m:sty m:val="p"></m:sty></m:scr></m:rpr>=</m:r><m:r>{1,...,</m:r><m:r>n</m:r><m:r>}</m:r></m:omath> be an <m:omath><m:r>n</m:r></m:omath>-elemen set and let symbol <m:omath><m:d><m:dpr><m:ctrlpr></m:ctrlpr></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><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:r>[</m:r><m:r>n</m:r><m:r>]</m:r></m:e></m:mr><m:mr><m:e><m:r>k</m:r></m:e></m:mr></m:m></m:e></m:d><m:r> </m:r></m:omath>denote the family of all k-element subsets (or k-sets) of <m:omath><m:r>[</m:r><m:r>n</m:r><m:r>]</m:r></m:omath>. A family <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></m:omath> of k-sets of <m:omath><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></m:omath> is said to be intersecting if for every two members A, B in <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></m:omath>, we have <m:omath><m:r>A</m:r><m:r>∩</m:r><m:r>B</m:r><m:r>≠∅</m:r></m:omath>.  For a fixed element <m:omath><m:r>i</m:r><m:r>∈[</m:r><m:r>n</m:r><m:r>]</m:r></m:omath>, if all members of <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></m:omath> contain i, then it is clear that <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></m:omath> is an intersecting family, which is called star. For each<m:omath><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></m:omath>, the family<m:omath><m:r> </m:r><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>S</m:r></m:e><m:sub><m:r>i</m:r></m:sub></m:ssub><m:r>={</m:r><m:r>A</m:r><m:r>:</m:r><m:d><m:dpr><m:begchr m:val="|"><m:endchr m:val="|"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>A</m:r></m:e></m:d><m:r>=</m:r><m:r>k</m:r><m:r>, </m:r><m:r>A</m:r><m:r>&sube;</m:r><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>n</m:r></m:e></m:d><m:r>, </m:r><m:r>i</m:r><m:r>∈</m:r><m:r>A</m:r></m:omath>} is a maximal star.   The well-known Erdős –Ko–Rado theorem is one of important results in extremal combinatorics. It has many interesting proofs and extensions. The Erdős-Ko-Rado theorem states that every intersecting family of <m:omath><m:d><m:dpr><m:ctrlpr></m:ctrlpr></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><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:r>[</m:r><m:r>n</m:r><m:r>]</m:r></m:e></m:mr><m:mr><m:e><m:r>k</m:r></m:e></m:mr></m:m></m:e></m:d></m:omath> has cardinality at most <m:omath><m:d><m:dpr><m:ctrlpr></m:ctrlpr></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><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e></m:mr><m:mr><m:e><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></m:e></m:mr></m:m></m:e></m:d></m:omath> provided that <m:omath><m:r>n</m:r><m:r>≥2</m:r><m:r>k</m:r></m:omath>; moreover, if <m:omath><m:r>n</m:r><m:r>>2</m:r><m:r>k</m:r></m:omath>, then  the only intersecting families  of this cardinality are isomorphic to <m:omath><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>S</m:r></m:e><m:sub><m:r>i</m:r></m:sub></m:ssub></m:omath>.   Two families <m:omath><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="p"></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>&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></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><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>n</m:r></m:e></m:d></m:e></m:mr><m:mr><m:e><m:r>k</m:r></m:e></m:mr></m:m></m:e></m:d></m:omath>  and <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></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><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>n</m:r></m:e></m:d></m:e></m:mr><m:mr><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></m:e></m:mr></m:m></m:e></m:d></m:omath> are called cross intersecting if for any <m:omath><m:r>A</m:r><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:omath> and <m:omath><m:r>B</m:r><m:r>∈</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></m:omath>, we have <m:omath><m:r>A</m:r><m:r>∩</m:r><m:r>B</m:r><m:r>≠∅</m:r></m:omath>.  Strengthening the  Erdős–Ko–Rado theorem, in 1986 Pyber  showed an upper bound for <m:omath><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="p"></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>||</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><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>|</m:r></m:omath> as follows. Theorem A. Let k, ℓ, n be positive integers such that k<m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>≥</m:r></m:omath>ℓ. Assume that <m:omath><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="p"></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>&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><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></m:e><m:e><m:r>k</m:r></m:e></m:eqarr></m:e></m:d></m:omath> and <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><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></m:e><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></m:e></m:eqarr></m:e></m:d></m:omath> are cross intersecting. <ul> <li style="margin-left:8px">If  k>ℓ and <m:omath><m:r>n</m:r><m:r>≥2</m:r><m:r>k</m:r><m:r>+</m:r><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></m:omath>, then <m:omath><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><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></m:e></m:eqarr></m:e></m:d><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><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></m:e></m:eqarr></m:e></m:d><m:r>.</m:r></m:omath></li> <li style="margin-left:8px">If k=ℓ and <m:omath><m:r>n</m:r><m:r>≥2</m:r><m:r>k</m:r></m:omath>, then <m:omath><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><m:ssup><m:ssuppr><m:ctrlpr></m:ctrlpr></m:ssuppr><m:e><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></m:e></m:eqarr></m:e></m:d></m:e><m:sup><m:r>2</m:r></m:sup></m:ssup><m:r>.</m:r></m:omath></li> </ul>    In 1989 Matsumoto and Tokushige slightly improved Pyber's result as follows. Theorem B. Let k, ℓ, n be positive integers such that <m:omath><m:r>n</m:r><m:r>≥2 </m:r><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>{</m:r><m:r>k</m:r></m:omath>, ℓ}. Assume that <m:omath><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="p"></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>&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><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></m:e><m:e><m:r>k</m:r></m:e></m:eqarr></m:e></m:d></m:omath> and <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><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></m:e><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></m:e></m:eqarr></m:e></m:d></m:omath> are cross intersecting. Then <m:omath><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><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></m:e></m:eqarr></m:e></m:d><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><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></m:e></m:eqarr></m:e></m:d><m:r>.</m:r></m:omath>   We say that a family <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></m:omath> is t-almost intersecting if for every set <m:omath><m:r>A</m:r><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:omath> there are at most t elements of <m:omath><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="p"></m:sty></m:scr></m:rpr> </m:r></m:omath>disjoint from <m:omath><m:r>A</m:r></m:omath>. In 2012 Gerbner, Lemons, Palmer, Patkós, and Szécsi proved an interesting generalization of the Erdős–Ko–Rado theorem for t-almost intersecting families. Theorem C. Let k, n. t be positive integers and <m:omath><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="p"></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>&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>n</m:r></m:e></m:d></m:e><m:e><m:r>k</m:r></m:e></m:eqarr></m:e></m:d><m:r> </m:r></m:omath>is a t-almost intersecting family. <ul> <li style="margin-left:8px">If  n=n(k,t) is sufficiently large, then <m:omath><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><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></m:e></m:eqarr></m:e></m:d></m:omath> with equality if and only if  <m:omath><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><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>S</m:r></m:e><m:sub><m:r>i</m:r></m:sub></m:ssub></m:omath> for some <m:omath><m:r>i</m:r><m:r>∈[</m:r><m:r>n</m:r><m:r>]</m:r></m:omath></li> <li style="margin-left:8px">If <m:omath><m:r> </m:r><m:r>k</m:r><m:r>≥3</m:r></m:omath>, <m:omath><m:r>n</m:r><m:r>≥2</m:r><m:r>k</m:r><m:r>+2</m:r></m:omath>, and <m:omath><m:r>t</m:r><m:r>=1</m:r></m:omath>, then <m:omath><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><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></m:e></m:eqarr></m:e></m:d></m:omath> with equality if and only if  <m:omath><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><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>S</m:r></m:e><m:sub><m:r>i</m:r></m:sub></m:ssub></m:omath> for some <m:omath><m:r>i</m:r><m:r>∈[</m:r><m:r>n</m:r></m:omath>].</li> </ul>  Main Results  We say two families <m:omath><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="p"></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>&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></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><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>n</m:r></m:e></m:d></m:e></m:mr><m:mr><m:e><m:r>k</m:r></m:e></m:mr></m:m></m:e></m:d></m:omath>  and <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></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><m:ctrlpr></m:ctrlpr></m:mpr><m:mr><m:e><m:d><m:dpr><m:begchr m:val="["><m:endchr m:val="]"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r>n</m:r></m:e></m:d></m:e></m:mr><m:mr><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></m:e></m:mr></m:m></m:e></m:d></m:omath> are cross t-almost intersecting if any  <m:omath><m:r>A</m:r><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:omath> is disjoint from at most t elements of <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B</m:r></m:omath> and any  <m:omath><m:r>B</m:r><m:r>∈</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></m:omath> is disjoint from at most t elements of<m:omath><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:omath>. As our main result we simultaneously extend the previous results for sufficiently large n. Theorem 1. Let k, ℓ, n be positive integers such that n=n (k, ℓ ,t) is sufficiently large. Assume that <m:omath><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="p"></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>&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><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></m:e><m:e><m:r>k</m:r></m:e></m:eqarr></m:e></m:d></m:omath> and <m:omath><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B&sube;</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><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></m:e><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>l</m:r></m:e></m:eqarr></m:e></m:d></m:omath> are cross t-almost intersecting. Then <m:omath><m:d><m:dpr><m:begchr m:val="|"><m:endchr m:val="|"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>A</m:r></m:e></m:d><m:d><m:dpr><m:begchr m:val="|"><m:endchr m:val="|"><m:ctrlpr></m:ctrlpr></m:endchr></m:begchr></m:dpr><m:e><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>B</m:r></m:e></m:d><m:r><m:rpr><m:scr m:val="script"><m:sty m:val="i"></m:sty></m:scr></m:rpr>≤</m:r><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><m:r>k</m:r><m:r>-</m:r><m:r>1</m:r></m:e></m:eqarr></m:e></m:d><m:d><m:dpr><m:ctrlpr></m:ctrlpr></m:dpr><m:e><m:eqarr><m:eqarrpr><m:ctrlpr></m:ctrlpr></m:eqarrpr><m:e><m:r>n</m:r><m:r>-</m:r><m:r>1</m:r></m:e><m:e><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></m:e></m:eqarr></m:e></m:d></m:omath> with equality if and only if  <m:omath><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><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><m:ssub><m:ssubpr><m:ctrlpr></m:ctrlpr></m:ssubpr><m:e><m:r>S</m:r></m:e><m:sub><m:r>i</m:r></m:sub></m:ssub></m:omath> for some  <m:omath><m:r>i</m:r><m:r>∈[</m:r><m:r>n</m:r></m:omath>].   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