Erdős asked if, for every pair of positive integers g and k, there exists a graph H having girth (H) = k and the property that every r-colouring of the edges of H yields a monochromatic cycle Ck. The existence of such graphs H was confirmed by the third author and Ruciński.We consider the related numerical problem of estimating the order of the smallest graph H with this property for given integers r and k. We show that there exists a graph H on R10k2; k15k3 vertices (where R = R(Ck; r) is the r-colour Ramsey number for the cycle Ck) having girth (H) = k and the Ramsey property that every r-colouring of the edges of H yields a monochromatic Ck Two related numerical problems regarding arithmetic progressions in subsets of the integers and cliques in graphs are also considered.

中文翻译：

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涉及大周长的图和超图的 Ramsey 型数

Erdős 询问是否对于每对正整数G和ķ, 存在一个图H有周长（H) =ķ以及每个人的财产r- 边缘的着色H产生单色循环Cķ. 此类图的存在H得到了第三作者和 Ruciński 的证实。我们考虑估计最小图的阶数的相关数值问题H对于给定的整数具有此属性r和ķ. 我们证明存在一个图H在R10ķ2;ķ15ķ3顶点（其中R=R(Cķ;r） 是个r- 循环的颜色 Ramsey 数Cķ) 有周长 (H) =ķ和拉姆齐财产，每个r- 边缘的着色H产生单色Cķ还考虑了关于图中整数子集和团的算术级数的两个相关数值问题。