标题： 关于模p下交集受限的集合系统上Alon-Babai-Suzuki猜想的加强不等式 显示英文标题

标题： A strengthened inequality of Alon-Babai-Suzuki's conjecture on set systems with restricted intersections modulo p

摘要： 设$K=\{k_1,k_2,\ldots,k_r\}$和$L=\{l_1,l_2,\ldots,l_s\}$是$\{0,1,\ldots,p-1\}$的不相交子集，其中$p$是一个素数且$A=\{A_1,A_2,\ldots,A_m\}$是$[n]$的子集族，使得$|A_i|\pmod{p}\in K$对所有$A_i\in A$和$|A_i\cap A_j|\pmod{p}\in L$对$i\ne j$。 1991年，Alon、Babai和Suzuki提出了一个猜想，即如果$n\geq s+\max_{1\leq i\leq r} k_i$，那么$|A|\leq {n\choose s}+{n\choose s-1}+\cdots+{n\choose s-r+1}$。 2000年，Qian和Ray-Chaudhuri在条件$n\geq 2s-r$下证明了该猜想。 2015年，Hwang和Kim验证了Alon、Babai和Suzuki的猜想。 在本文中，我们将证明，如果$n\geq 2s-2r+1$或$n\geq s+\max_{1\leq i\leq r}k_i$，那么\[ |A|\leq{n-1\choose s}+{n-1\choose s-1}+\cdots+{n-1\choose s-2r+1}. \]。这一结果在$n\geq 2s-2$时加强了Alon、Babai和Suzuki猜想的上界。 显示英文摘要

摘要： Let $K=\{k_1,k_2,\ldots,k_r\}$ and $L=\{l_1,l_2,\ldots,l_s\}$ be disjoint subsets of $\{0,1,\ldots,p-1\}$, where $p$ is a prime and $A=\{A_1,A_2,\ldots,A_m\}$ be a family of subsets of $[n]$ such that $|A_i|\pmod{p}\in K$ for all $A_i\in A$ and $|A_i\cap A_j|\pmod{p}\in L$ for $i\ne j$. In 1991, Alon, Babai and Suzuki conjectured that if $n\geq s+\max_{1\leq i\leq r} k_i$, then $|A|\leq {n\choose s}+{n\choose s-1}+\cdots+{n\choose s-r+1}$. In 2000, Qian and Ray-Chaudhuri proved the conjecture under the condition $n\geq 2s-r$. In 2015, Hwang and Kim verified the conjecture of Alon, Babai and Suzuki. In this paper, we will prove that if $n\geq 2s-2r+1$ or $n\geq s+\max_{1\leq i\leq r}k_i$, then \[ |A|\leq{n-1\choose s}+{n-1\choose s-1}+\cdots+{n-1\choose s-2r+1}. \] This result strengthens the upper bound of Alon, Babai and Suzuki's conjecture when $n\geq 2s-2$.