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arXiv · 2310.20461

Ramsey numbers of bounded degree trees versus general graphs

Richard Montgomery·Matías Pavez-Signé·Jun Yan
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Abstract

For every $k\ge 2$ and $Δ$, we prove that there exists a constant $C_{Δ,k}$ such that the following holds. For every graph $H$ with $χ(H)=k$ and every tree with at least $C_{Δ,k}|H|$ vertices and maximum degree at most $Δ$, the Ramsey number $R(T,H)$ is $(k-1)(|T|-1)+σ(H)$, where $σ(H)$ is the size of a smallest colour class across all proper $k$-colourings of $H$. This is tight up to the value of $C_{Δ,k}$, and confirms a conjecture of Balla, Pokrovskiy, and Sudakov.

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Richard Montgomery, Matías Pavez-Signé, Jun Yan. 2023-10-31. Ramsey numbers of bounded degree trees versus general graphs. https://doi.org/10.1016/j.jctb.2025.02.004

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