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arXiv · 2207.05048

Size-Ramsey numbers of graphs with maximum degree three

Nemanja Draganić·Kalina Petrova
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Abstract

The size-Ramsey number $\hat{r}(H)$ of a graph $H$ is the smallest number of edges a (host) graph $G$ can have, such that for any red/blue colouring of $G$, there is a monochromatic copy of $H$ in $G$. Recently, Conlon, Nenadov and Truji\'c showed that if $H$ is a graph on $n$ vertices and maximum degree three, then $\hat{r}(H) = O(n^{8/5})$, improving upon the upper bound of $n^{5/3 + o(1)}$ by Kohayakawa, R\"odl, Schacht and Szemer\'edi. In this paper we show that $\hat{r}(H)\leq n^{3/2+o(1)}$. While the previously used host graphs were vanilla binomial random graphs, we prove our result using a novel host graph construction. Our bound hits a natural barrier of the existing methods.

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Nemanja Draganić, Kalina Petrova. 2022-07-11. Size-Ramsey numbers of graphs with maximum degree three. https://arxiv.org/abs/2207.05048

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