K4-intersecting families of graphs

Ellis, Filmus, and Friedgut proved an old conjecture of Simonovits and Sos 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 is an element of {3, 4}, showing that if F1 and F2 are families of graphs on n labeled vertices such that for any G1 is an element of F1 and G2 is an element of F2, G1 boolean AND 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 boolean AND G2 contains a Kt" can be replaced by "G1 and G2 agree on a non-(t - 1) colorable graph."(c) 2023 Elsevier Inc. All rights reserved.