K4-intersecting families of graphs

Abstract

Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and Sós showing that any triangle-intersecting family of graphs on n vertices has size at most 2(n2)−3, with equality for the family of graphs containing some fixed triangle. They conjectured that their results extend to cross-intersecting families, as well to Kt-intersecting families. We prove these conjectures for t∈{3,4}, showing that if F1 and F2 are families of graphs on n labeled vertices such that for any G1∈F1 and G2∈F2, G1∩G2 contains a Kt, then |F1||F2|≤4(n2)−(t2), with equality if and only if F1=F2 consists of all graphs that contain some fixed Kt. We also establish a stability result. More generally, “G1∩G2 contains a Kt” can be replaced by “G1 and G2 agree on a non-(t−1)-colorable graph.”