Anti-Ramsey Multiplicities
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 commo...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.1801.00474 |