Multi-part cross-intersecting families

Let \({\mathcal {A}}\subseteq {[n]\atopwithdelims ()a}\) and \({\mathcal {B}}\subseteq {[n]\atopwithdelims ()b}\) be two families of subsets of [n], we say \({\mathcal {A}}\) and \({\mathcal {B}}\) are cross-intersecting if \(A\cap B\ne \emptyset \) for all \(A\in {\mathcal {A}}\), \(B\in {\mathcal {B}}\). 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.