Induced Ramsey problems for trees and graphs with bounded treewidth
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866910467554803712 |
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| author | Hunter, Zach Sudakov, Benny |
| author_facet | Hunter, Zach Sudakov, Benny |
| contents | The induced $q$-color size-Ramsey number $\hat{r}_{\text{ind}}(H;q)$ of a graph $H$ is the minimal number of edges a host graph $G$ can have so that every $q$-edge-coloring of $G$ contains a monochromatic copy of $H$ which is an induced subgraph of $G$. A natural question, which in the non-induced case has a very long history, asks which families of graphs $H$ have induced Ramsey numbers that are linear in $|H|$. We prove that for every $k,w,q$, if $H$ is an $n$-vertex graph with maximum degree $k$ and treewidth at most $w$, then $\hat{r}_{\text{ind}}(H;q) = O_{k,w,q}(n)$. This extends several old and recent results in Ramsey theory. Our proof is quite simple and relies upon a novel reduction argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_00352 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Induced Ramsey problems for trees and graphs with bounded treewidth Hunter, Zach Sudakov, Benny Combinatorics The induced $q$-color size-Ramsey number $\hat{r}_{\text{ind}}(H;q)$ of a graph $H$ is the minimal number of edges a host graph $G$ can have so that every $q$-edge-coloring of $G$ contains a monochromatic copy of $H$ which is an induced subgraph of $G$. A natural question, which in the non-induced case has a very long history, asks which families of graphs $H$ have induced Ramsey numbers that are linear in $|H|$. We prove that for every $k,w,q$, if $H$ is an $n$-vertex graph with maximum degree $k$ and treewidth at most $w$, then $\hat{r}_{\text{ind}}(H;q) = O_{k,w,q}(n)$. This extends several old and recent results in Ramsey theory. Our proof is quite simple and relies upon a novel reduction argument. |
| title | Induced Ramsey problems for trees and graphs with bounded treewidth |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.00352 |