Abstract
Let \(ex(n, C_{\le m})\) denote the maximum size of a graph of order n and girth at least \(m+1\), and \(EX(n, C_{\le m})\) be the set of all graphs of girth at least \(m+1\) and size \(ex(n, C_{\le m})\). The Ramsey number \(R_l(C_{\le m})\) is the smallest n such that every \(K_n\), whose edges are in l colors, must contain a monochromatic cycle of length k for some \(3\le k\le m\). In this paper, we study the exact values of \(R_l(C_{\le l+1})\). By using the known results of \(ex(n, C_{\le l+1})\), we first give the upper bounds on \(R_l(C_{\le l+1})\), then we prove that \(R_l(C_{\le l+1})=2l+3\) for odd \(l\ge 3\). For even l, we prove that \(R_4(C_{\le 5})=12\), \(R_6(C_{\le 7})=16\), and \(R_l(C_{\le l+1})=2l+3\) for \(8\le l\le 12\), leaving the case of \(l\ge 14\) open.
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The authors are grateful to the referees and Professor Stanisław P. Radziszowski for their valuable comments and suggestions which led to the improvement of the present version.
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Supported by NSFC (61572005, 61272004, 61672086, 61702030), and Fundamental Research Funds for Central Universities.
Appendix: The Matrices of \(K_{18}\), \(K_{22}\) and \(K_{26}\)
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Zhu, W., Sun, Y., Wu, Y. et al. Exact Values of Multicolor Ramsey Numbers \(R_l(C_{\le l+1})\). Graphs and Combinatorics 36, 839–852 (2020). https://doi.org/10.1007/s00373-020-02157-w
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DOI: https://doi.org/10.1007/s00373-020-02157-w