On Cross-intersecting Families of Sets

A product version of the Erdős–Ko–Rado theorem

“…, Some values of m × (n, k, r, t) are known. For example, Pyber [20], Matsumoto and the author [16], and Bey [2] proved the following:…”

“…Indeed, we will consider two types of problems related to (2); one is about k-uniform cross intersecting families and the other is about the p-weight of cross intersecting families.…”

On cross t-intersecting families of sets

“…Two families F , G ⊂ 2 [n] are called cross t-intersecting if |F ∩ G| t holds for all F ∈ F , G ∈ G . Pyber [11] generalized the Erdős-Ko-Rado theorem [5] to cross 1-intersecting families, and the result was slightly refined by Matsumoto and Tokushige [9] and Bey [2] as follows. For a real p ∈ (0, 1) and a family G ⊂ 2 [n] we define the p-weight of G , denoted by w p (G ), as follows:…”

“…Fix y := x + ε 1 ∈ Y . Using j k this is equivalent to j · · · (k + 1) (z − n + j) · · · (z − n + k + 1), which follows from z n. 2 …”

An Analogue of the Hilton-Milner Theorem for Weak Compositions

“…In the celebrated paper [1], Ahlswede and Khachatrian extended the Erdős-Ko-Rado theorem by determining the structure of all t-intersecting set systems of maximum size for all possible n (see also [3,17,25,29,37,39,40,41] for some related results). There have been many recent results showing that a version of the Erdős-Ko-Rado theorem holds for combinatorial objects other than set systems.…”