Set systems with \(\mathcal L\)-intersections modulo a prime number
Let \(p\) be a prime, \(L= \{l_1,l_2,\dots, l_s\}\) and \(K= \{k_1,k_2,\dots, k_r\}\) be two disjoint subsets of \(\{0,1,2,\dots, p-1\}\). Suppose \(F= \{F_1,F_2,\dots, F_m\}\) is a family of subsets of \(\{1,2,\dots, n\}\) such that \(|F_i\cap F_j|\pmod p\in L\) for every pair \(i\neq j\) and \(|F_i|\pmod p\in K\) for every \(1\leq i\leq m\). In [Combinatorica 23, No. 3, 527--533 (2003; Zbl 1045.05087)] \textit{H. S. Snevily} conjectured that \(|F|\leq{n\choose s}\). In this paper the authors show that if \(F\) is a \(k\)-uniform family of subsets of \(\{1,2,\dots, n\}\), then the above-mentioned conjecture is true.
scientific article; zbMATH DE number 5523120