Ramsey size linear and generalization
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917368416960512 |
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| author | Hng, Eng Keat Ji, Meng Lamaison, Ander |
| author_facet | Hng, Eng Keat Ji, Meng Lamaison, Ander |
| contents | More than thirty years ago, Erdős, Faudree, Rousseau, and Schelp posed a fundamental question in extremal graph theory: What is the optimal constant $c_k$ such that $r(C_{2k+1}, G) \le c_k m$ for any graph $G$ with $m$ edges and no isolated vertices? In this paper, we make a significant step towards answering this question by proving that $r(C_{2k+1}, G) \le (2 + o(1)) m + p,$ where $p$ denotes the number of vertices in $G$. Additionally, we extend the work of Goddard and Kleitman and independently Sidorenko, who proved that $r(K_3, G) \le 2m + 1$ for any graph $G$ with $m$ edges and no isolated vertices. We generalize their findings to the clique version, establishing that $r(K_r, G) \le c_r m^{(r-1)/2}$, and to the multicolor setting, showing that $r_{k+1}(K_3; G) \le c_k m^{(k+1)/2}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_25453 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ramsey size linear and generalization Hng, Eng Keat Ji, Meng Lamaison, Ander Combinatorics More than thirty years ago, Erdős, Faudree, Rousseau, and Schelp posed a fundamental question in extremal graph theory: What is the optimal constant $c_k$ such that $r(C_{2k+1}, G) \le c_k m$ for any graph $G$ with $m$ edges and no isolated vertices? In this paper, we make a significant step towards answering this question by proving that $r(C_{2k+1}, G) \le (2 + o(1)) m + p,$ where $p$ denotes the number of vertices in $G$. Additionally, we extend the work of Goddard and Kleitman and independently Sidorenko, who proved that $r(K_3, G) \le 2m + 1$ for any graph $G$ with $m$ edges and no isolated vertices. We generalize their findings to the clique version, establishing that $r(K_r, G) \le c_r m^{(r-1)/2}$, and to the multicolor setting, showing that $r_{k+1}(K_3; G) \le c_k m^{(k+1)/2}.$ |
| title | Ramsey size linear and generalization |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.25453 |