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arXiv · 2410.08644

Canonical Ramsey numbers of sparse graphs

Lior Gishboliner·Aleksa Milojević·Benny Sudakov·Yuval Wigderson
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Abstract

The canonical Ramsey theorem of Erd\H{o}s and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy of $H$. The least such $N$ is called the Erd\H{o}s-Rado number of $H$, denoted by $ER(H)$. Erd\H{o}s-Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erd\H{o}s-Rado numbers of sparse graphs. For example, we prove that if $H$ has bounded degree, then $ER(H)$ is polynomial in $|V(H)|$ if $H$ is bipartite, but exponential in general. We also study the closely-related problem of constrained Ramsey numbers. For a given tree $S$ and given path $P_t$, we study the minimum $N$ such that every edge-coloring of $K_N$ contains a monochromatic copy of $S$ or a rainbow copy of $P_t$. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse-Ackermann type.

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Lior Gishboliner, Aleksa Milojević, Benny Sudakov, Yuval Wigderson. 2024-10-11. Canonical Ramsey numbers of sparse graphs. https://arxiv.org/abs/2410.08644

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