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arXiv · 2606.02389

A degree version of the Burr-Erd\H{o}s conjecture on trees

Jasmin Katz·Matías Pavez-Signé·Jozef Skokan
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Abstract

An old conjecture of Burr and Erd\H os states that the Ramsey number of any $n$-vertex tree $T$ is at most $2n-2$. In 2012, Schelp asked whether a degree version of the Burr--Erd\H{o}s conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $\Delta\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $\delta(G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $\Delta$. We prove this conjecture in a strong form, showing that it is true even if one removes the extra $\varepsilon n$ term in the size of the host graph.

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Jasmin Katz, Matías Pavez-Signé, Jozef Skokan. 2026-06-01. A degree version of the Burr-Erd\H{o}s conjecture on trees. https://arxiv.org/abs/2606.02389

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