### References & Citations

# Mathematics > Combinatorics

# Title: Anti-Ramsey Multiplicities

(Submitted on 1 Jan 2018 (this version),

*latest version 28 Mar 2018*(v2))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.

## Submission history

From: Jessica De Silva [view email]**[v1]**Mon, 1 Jan 2018 16:55:10 GMT (12kb)

**[v2]**Wed, 28 Mar 2018 02:43:54 GMT (11kb)

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