Canonical Ramsey numbers of sparse graphs
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912069112037376 |
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| author | Gishboliner, Lior Milojević, Aleksa Sudakov, Benny Wigderson, Yuval |
| author_facet | Gishboliner, Lior Milojević, Aleksa Sudakov, Benny Wigderson, Yuval |
| contents | The canonical Ramsey theorem of Erdős and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy of $H$. The least such $N$ is called the Erdős-Rado number of $H$, denoted by $ER(H)$. Erdős-Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erdős-Rado numbers of sparse graphs. For example, we prove that if $H$ has bounded degree, then $ER(H)$ is polynomial in $|V(H)|$ if $H$ is bipartite, but exponential in general.
We also study the closely-related problem of constrained Ramsey numbers. For a given tree $S$ and given path $P_t$, we study the minimum $N$ such that every edge-coloring of $K_N$ contains a monochromatic copy of $S$ or a rainbow copy of $P_t$. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse-Ackermann type. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_08644 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Canonical Ramsey numbers of sparse graphs Gishboliner, Lior Milojević, Aleksa Sudakov, Benny Wigderson, Yuval Combinatorics 05D10 The canonical Ramsey theorem of Erdős and Rado implies that for any graph $H$, any edge-coloring (with an arbitrary number of colors) of a sufficiently large complete graph $K_N$ contains a monochromatic, lexicographic, or rainbow copy of $H$. The least such $N$ is called the Erdős-Rado number of $H$, denoted by $ER(H)$. Erdős-Rado numbers of cliques have received considerable attention, and in this paper we extend this line of research by studying Erdős-Rado numbers of sparse graphs. For example, we prove that if $H$ has bounded degree, then $ER(H)$ is polynomial in $|V(H)|$ if $H$ is bipartite, but exponential in general. We also study the closely-related problem of constrained Ramsey numbers. For a given tree $S$ and given path $P_t$, we study the minimum $N$ such that every edge-coloring of $K_N$ contains a monochromatic copy of $S$ or a rainbow copy of $P_t$. We prove a nearly optimal upper bound for this problem, which differs from the best known lower bound by a function of inverse-Ackermann type. |
| title | Canonical Ramsey numbers of sparse graphs |
| topic | Combinatorics 05D10 |
| url | https://arxiv.org/abs/2410.08644 |