Anti-Ramsey Multiplicities

TL;DR

This paper explores the maximum number of rainbow subgraphs in edge-colored complete graphs, introducing the concept of anti-common graphs and showing that not all bipartite graphs are anti-common, contrasting with the monochromatic case.

Contribution

It investigates anti-Ramsey multiplicity for various graph families and identifies classes that are anti-common or not, challenging existing conjectures.

Findings

Not all bipartite graphs are anti-common.

Some graph classes behave differently in rainbow settings.

The rainbow Sidorenko's conjecture is false.

Abstract

The Ramsey multiplicity constant of a graph $H$ is the minimum proportion of copies of $H$ in the complete graph which are monochromatic under an edge-coloring of $K_n$ as $n$ goes to infinity. Graphs for which this minimum is asymptotically achieved by taking a random coloring are called {\em common}, and common graphs have been studied extensively, leading to the Burr-Rosta conjecture and Sidorenko's conjecture. Erd\H{o}s and S\'os asked what the maximum number of rainbow triangles is in a $3$-coloring of the edge set of $K_n$, a rainbow version of the Ramsey multiplicity question. A graph $H$ is called $r$-anti-common if the maximum proportion of rainbow copies of $H$ in any $r$-coloring of $E(K_n)$ is asymptotically achieved by taking a random coloring. In this paper, we investigate anti-Ramsey multiplicity for several families of graphs. We determine classes of graphs which are either anti-common or not. Some of these classes follow the same behavior as the monochromatic case, but some of them do not. In particular the rainbow equivalent of Sidorenko's conjecture, that all bipartite graphs are anti-common, is false.