A rainbow version of Mantel's Theorem

Abstract

Mantel's Theorem asserts that a simple $n$ vertex graph with more than $\frac{1}{4}{n}^2$ edges has a triangle (three mutually adjacent vertices). Here we consider a rainbow variant of this problem. We prove that whenever $G_1,G_2,G_3$ are simple graphs on a common set of $n$ vertices and $|E(G_i)|>(\frac{26-2\sqrt{7}}{81})n^2\approx 0.2557n^2$ for $1\le i\le 3$, then there exist distinct vertices $v_1,v_2,v_3$ so that (working with the indices modulo 3) we have $v_i{v}_{i+1}\in E(G_i)$ for $1\le i\le 3$. We provide an example to show this bound is best possible. This also answers a question of Diwan and Mubayi. We include a new short proof of Mantel's Theorem we obtained as a byproduct.

12 pages, 3 figures