Some remarks on graphs without rainbow triangles

Ervin Győri, Zhen He, Zequn Lv, Nika Salia, Casey Tompkins, Kitti Varga, Xiutao Zhu

Published 2022-04-15Version 1

Given graphs $G_1$, $G_2$ and $G_3$ on a common vertex set of size $n$, a rainbow triangle is a triangle consisting of one edge from each $G_i$. In this note we provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. Moreover we propose an alternative conjecture and prove it in the case when every pair is an edge in at least one of the graphs.