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.

令\ {ex（n，C _ {\ le m}）\）表示n阶和周长图的最大尺寸至少为\（m + 1 \）和\（EX（n，C _ {\ le m }）\）至少为\（m + 1 \）和大小\（ex（n，C _ {\ le m}）\）的所有周长图的集合。拉姆齐数\（R_L（C _ {\文件米}）\）是最小Ñ使得每个\（K_n \） ，其边缘是在升的颜色，必须包含长度的单色周期ķ一些\（3 \ le k \ le m \）。在本文中，我们研究\（R_l（C _ {\ le l + 1}）\）的精确值。通过使用已知的结果\（ex（n，C _ {\ le l + 1}）\），我们首先给出\（R_l（C _ {\ le l + 1}）\）的上限，然后证明\（R_l（C_ {\ le l + 1}）= 2l + 3 \）表示奇数\（l \ ge 3 \）。对于偶数l，我们证明\（R_4（C _ {\ le 5}）= 12 \），\（R_6（C _ {\ le 7}）= 16 \）和\（R_l（C _ {\ le l + 1}）= 2l + 3 \）表示\（8 \ le l \ le 12 \），打开\（l \ ge 14 \）的情况。