A degree version of the Burr-Erdős conjecture on trees
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918535269187584 |
|---|---|
| author | Katz, Jasmin Pavez-Signé, Matías Skokan, Jozef |
| author_facet | Katz, Jasmin Pavez-Signé, Matías Skokan, Jozef |
| contents | 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--Erdős conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $Δ\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $δ(G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $Δ$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2606_02389 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A degree version of the Burr-Erdős conjecture on trees Katz, Jasmin Pavez-Signé, Matías Skokan, Jozef Combinatorics 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--Erdős conjecture holds. More precisely, Schelp asked if is it true that for any $\varepsilon>0$ and $Δ\ge 2$, if $G$ is a graph on $N\ge (2+\varepsilon)n$ vertices and minimum degree $δ(G)\ge \lfloor 3N/4\rfloor$, then every blue/red colouring of the edges of $G$ yields a monochromatic copy of each $n$-vertex tree with maximum degree at most $Δ$. 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. |
| title | A degree version of the Burr-Erdős conjecture on trees |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2606.02389 |