Density of rainbow triangles and properly colored $K_4$'s

TL;DR

This paper establishes sharp upper bounds on the number of rainbow triangles and properly colored $K_4$'s in edge-colored graphs, using novel flag-algebra, counting, and entropy methods, with proofs avoiding computers.

Contribution

It introduces new flag-algebra and entropy proofs for bounds on rainbow subgraphs, including a novel version of H"older's inequality for these problems.

Findings

Sharp upper bounds on rainbow triangles in colored graphs.

New flag-algebra and entropy proofs avoiding computational methods.

A novel flag-algebra version of H"older's inequality.

Abstract

We establish a sharp upper bound on the number of properly $3$-edge-colored $K_4$'s in graphs with $R$ red, $G$ green and $B$ blue edges. We give a computer-free flag-algebra proof of this bound, and we also convert our proof into a classical counting proof and an entropy proof. Additionally, for every $k\ge 4$, for a fixed rainbow coloring $F$ of a complete graph $K_k$, we give a sharp upper bound on the number copies of $F$ in a $\binom{k}{2}$-edge-colored graph. Our proof of this result relies on a new flag-algebra version of H\"older's inequality. We also give a computer-free flag-algebra proof of the fact that a graph with $R$ red, $G$ green, and $B$ blue edges has at most $\sqrt{2 RGB}$ rainbow triangles, which was originally proven by T.-W. Chao and H.-H. H. Yu using the entropy method. We also provide an even shorter entropy proof of their result.