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

Abstract

Let $K=\{k_1,k_2,\dots ,k_r\}$ and $L=\{l_1,l_2,\dots ,l_s\}$ be disjoint subsets of $\{0,1,\dots ,p-1\}$, where $p$ is a prime and $A=\{A_1,A_2,\dots ,A_m\}$ be a family of subsets of $[n]$ such that $|A_i|(\mathrm{mod}p)\in K$ for all $A_i\in A$ and $|A_i\cap A_j|(\mathrm{mod}p)\in L$ for $i\ne j$. In 1991, Alon, Babai and Suzuki conjectured that if $n\ge s+{\max }_{1\le i\le r}{k}_i$, then $|A|\le \left(\genfrac{}{}{0ex}{}{n}{s}\right)+\left(\genfrac{}{}{0ex}{}{n}{s-1}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{n}{s-r+1}\right)$. In 2000, Qian and Ray-Chaudhuri proved the conjecture under the condition $n\ge 2s-r$. In 2015, Hwang and Kim verified the conjecture of Alon, Babai and Suzuki. In this paper, we will prove that if $n\ge 2s-2r+1$ or $n\ge s+{\max }_{1\le i\le r}{k}_i$, 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 $n\ge 2s-2$.