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arXiv · 2108.00987

Threshold Ramsey multiplicity for odd cycles

David Conlon·Jacob Fox·Benny Sudakov·Fan Wei
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Abstract

The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices.

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David Conlon, Jacob Fox, Benny Sudakov, Fan Wei. 2021-08-02. Threshold Ramsey multiplicity for odd cycles. https://arxiv.org/abs/2108.00987

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