Ramsey number of a cycle versus a graph of a given size

Stijn Cambie; Andrea Freschi; Patryk Morawski; Kalina Petrova; Alexey Pokrovskiy

Ramsey number of a cycle versus a graph of a given size

Combinatorics 2026-01-16 v1

Authors: Stijn Cambie , Andrea Freschi , Patryk Morawski , Kalina Petrova , Alexey Pokrovskiy

Abstract

In this paper, we prove that for every $k$ and every graph $H$ with $m$ edges and no isolated vertices, the Ramsey number $R(C_k,H)$ is at most $2m+\lfloor \frac{k-1}{2} \rfloor$ , provided $m$ is sufficiently large with respect to $k$ . This settles a problem of Erd\H{o}s, Faudree, Rousseau and Schelp.

Keywords

Ramsey number cycle graph isolated vertices Erdős-Faudree-Rousseau-Schelp problem

Cite

@article{arxiv.2601.10238,
  title  = {Ramsey number of a cycle versus a graph of a given size},
  author = {Stijn Cambie and Andrea Freschi and Patryk Morawski and Kalina Petrova and Alexey Pokrovskiy},
  journal= {arXiv preprint arXiv:2601.10238},
  year   = {2026}
}

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8 pages, 2 figures

Ramsey number of a cycle versus a graph of a given size

Abstract

Keywords

Cite

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