Ramsey numbers upon vertex deletion
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866911758671675392 |
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| author | Wigderson, Yuval |
| author_facet | Wigderson, Yuval |
| contents | Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor.
One consequence of this result is the following. There exists a family of graphs $\{G_n\}$ so that in any Ramsey coloring for $G_n$ (that is, a coloring of a clique on $r(G_n)-1$ vertices with no monochromatic copy of $G_n$), one of the color classes has density $o(1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_11181 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Ramsey numbers upon vertex deletion Wigderson, Yuval Combinatorics Given a graph $G$, its Ramsey number $r(G)$ is the minimum $N$ so that every two-coloring of $E(K_N)$ contains a monochromatic copy of $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from $G$, the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs $\{G_n\}$ so that in any Ramsey coloring for $G_n$ (that is, a coloring of a clique on $r(G_n)-1$ vertices with no monochromatic copy of $G_n$), one of the color classes has density $o(1)$. |
| title | Ramsey numbers upon vertex deletion |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2208.11181 |