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- W4385245538 abstract "Size-Ramsey numbers are a central notion in combinatorics and have been widely studied since their introduction by ErdH{o}s, Faudree, Rousseau and Schelp in 1978. Research has mainly focused on the size-Ramsey numbers of $n$-vertex graphs with constant maximum degree $Delta$. For example, graphs which also have constant treewidth are known to have linear size-Ramsey numbers. On the other extreme, the canonical examples of graphs of unbounded treewidth are the grid graphs, for which the best known bound has only very recently been improved from $O(n^{3/2})$ to $O(n^{5/4})$ by Conlon, Nenadov and Truji'c. In this paper, we prove a common generalization of these results by establishing new bounds on the size-Ramsey numbers in terms of treewidth (which may grow as a function of $n$). As a special case, this yields a bound of $tilde{O}(n^{3/2 - 1/2Delta})$ for proper minor-closed classes of graphs. In particular, this bound applies to planar graphs, addressing a question of Kamcev, Liebenau, Wood and Yepremyan. Our proof combines methods from structural graph theory and classic Ramsey-theoretic embedding techniques, taking advantage of the product structure exhibited by graphs with bounded treewidth." @default.
- W4385245538 created "2023-07-26" @default.
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- W4385245538 date "2023-07-22" @default.
- W4385245538 modified "2023-10-14" @default.
- W4385245538 title "Size-Ramsey numbers of structurally sparse graphs" @default.
- W4385245538 doi "https://doi.org/10.48550/arxiv.2307.12028" @default.
- W4385245538 hasPublicationYear "2023" @default.
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