Abstrakt
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 $|E(G_i)| > ( \frac{ 26 - 2 \sqrt{7} }{81})n^2 \approx 0.2557 n^2$ 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.