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arXiv · 2109.00734

Ramsey Numbers of Trails

Masatoshi Osumi
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Abstract

We initiate the study of Ramsey numbers of trails. Let $k \geq 2$ be a positive integer. The Ramsey number of trails with $k$ vertices is defined as the the smallest number $n$ such that for every graph $H$ with $n$ vertices, $H$ or the complete $\overline{H}$ contains a trail with $k$ vertices. We prove that the Ramsey number of trails with $k$ vertices is at most $k$ and at least $2\sqrt{k}+\Theta(1)$. This improves the trivial upper bound of $\lfloor 3k/2\rfloor -1$.

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Masatoshi Osumi. 2021-09-02. Ramsey Numbers of Trails. https://doi.org/10.1587/transfun.2021dmp0003

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