Alon–Babai–Suzuki’s inequalities, Frankl–Wilson type theorem and multilinear polynomials

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• 作者：Gyoyong Sohn^a ; Cheon Seoung Ryoo^b ; Philsu Kim^c ; Kyung-Won Hwang^d ; ^{khwang@dau.ac.kr"" rel=""nofollow} ; ^{khwang@donga.ac.kr"" rel=""nofollow} ; Jinsoo Hwang^e
• 关键词：Alon&#x2013 ; Babai&#x2013 ; Suzuki&#x2019 ; s inequalities ; Frankl&#x2013 ; Wilson theorem ; Multilinear polynomials
• 刊名：Applied Mathematics Letters
• 出版年：2011
• 出版时间：September 2011
• 年：2011
• 卷：24
• 期：9
• 页码：1477-1480
• 全文大小：218 K

文摘

Let K={k₁,k₂,…,k_r} and L={l₁,l₂,…,l_s} be subsets of {0,1,…,p−1} such that K∩L=0/, where p is a prime. Let be a family of subsets of [n]={1,2,…,n} with F_i () K for all and F_i∩F_j () L for any i≠j. Every subset F_i of [n] can be represented by a binary code such that a_j=1 if jF_i and a_j=0 if jF_i. Alon–Babai–Suzuki proved in non-modular version that if k_i≥s−r+1 for all i, then . We generalize it in modular version. Alon–Babai–Suzuki also proved that the above bound still holds under r(s−r+1)≤p−1 and in modular version. Alon–Babai–Suzuki made a conjecture that if they drop one condition r(s−r+1)≤p−1 among r(s−r+1)≤p−1 and , then the above bound holds. But we prove the same bound under dropping the opposite condition . So we prove the same bound under only condition r(s−r+1)≤p−1. This is a generalization of Frankl–Wilson theorem (Frankl and Wilson, 1981 [2]).