Ramsey sequences with bounded clique size
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866916974833958912 |
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| author | Aher, Abhishek Girish S, Aparna Lakshmanan |
| author_facet | Aher, Abhishek Girish S, Aparna Lakshmanan |
| contents | A sequence of graphs $ \{G_k\} $ is a Ramsey sequence if for every positive integer $ k $, the graph $ G_k $ is a proper subgraph of $ G_{k+1} $, and there exists an integer $n > k$ such that every red-blue coloring of $ G_n $ contains a monochromatic copy of $ G_k $. Among the wide range of open problems in Ramsey theory, an interesting open question is ``Does there exist an ascending sequence $\{G_k\}$ with $\lim_{k \to \infty} χ(G_k) = \infty$ and $\lim_{k \to \infty} ω(G_k) \neq \infty$ that is a Ramsey sequence?". In this paper, we solve this problem by constructing a Ramsey sequence $\{G_k\}$ with a bounded clique number such that $\lim_{k \to \infty} χ(G_k) = \infty$. Furthermore, using the observation that any monotonic increasing sequence of graphs that contains a Ramsey sequence as a subgraph is also Ramsey, we can generate infinitely many Ramsey sequences using this example. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23929 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ramsey sequences with bounded clique size Aher, Abhishek Girish S, Aparna Lakshmanan Combinatorics 05C35, 05C55, 05C15 A sequence of graphs $ \{G_k\} $ is a Ramsey sequence if for every positive integer $ k $, the graph $ G_k $ is a proper subgraph of $ G_{k+1} $, and there exists an integer $n > k$ such that every red-blue coloring of $ G_n $ contains a monochromatic copy of $ G_k $. Among the wide range of open problems in Ramsey theory, an interesting open question is ``Does there exist an ascending sequence $\{G_k\}$ with $\lim_{k \to \infty} χ(G_k) = \infty$ and $\lim_{k \to \infty} ω(G_k) \neq \infty$ that is a Ramsey sequence?". In this paper, we solve this problem by constructing a Ramsey sequence $\{G_k\}$ with a bounded clique number such that $\lim_{k \to \infty} χ(G_k) = \infty$. Furthermore, using the observation that any monotonic increasing sequence of graphs that contains a Ramsey sequence as a subgraph is also Ramsey, we can generate infinitely many Ramsey sequences using this example. |
| title | Ramsey sequences with bounded clique size |
| topic | Combinatorics 05C35, 05C55, 05C15 |
| url | https://arxiv.org/abs/2509.23929 |