Multi-part cross-intersecting families

Abstract

AbstractLet $${\mathcal {A}}\subseteq {[n]\atopwithdelims ()a}$$ A ⊆ [ n ] a and $${\mathcal {B}}\subseteq {[n]\atopwithdelims ()b}$$ B ⊆ [ n ] b be two families of subsets of [n], we say $${\mathcal {A}}$$ A and $${\mathcal {B}}$$ B are cross-intersecting if $$A\cap B\ne \emptyset $$ A ∩ B ≠ ∅ for all $$A\in {\mathcal {A}}$$ A ∈ A , $$B\in {\mathcal {B}}$$ B ∈ B . In this paper, we study cross-intersecting families in the multi-part setting. By characterizing the independent sets of vertex-transitive graphs and their direct products, we determine the sizes and structures of maximum-sized multi-part cross-intersecting families. This generalizes the results of Hilton’s (J Lond Math Soc 15(2):369–376, 1977) and Frankl–Tohushige’s (J Comb Theory Ser A 61(1):87–97, 1992) on cross-intersecting families in the single-part setting.