For graphs G and H, let \(G\xrightarrow {k}H\) signify that any k-edge coloring of G contains a monochromatic H as a subgraph. Let \(\mathcal{G}(K_2(N),p)\) be random graph spaces with edge probability p, where \(K_2(N)\) is the complete \(N\times N\) bipartite graph. For a constant \(p\in (0,1]\) and a cycle \(C_{2n}\) of length 2n, it is shown that \(\Pr [\mathcal{G}(K_2((k+o(1))n),p)\xrightarrow {k} C_{2n}]\rightarrow 1\) as \(n\rightarrow \infty \), where \(k=2,3\).

中文翻译：

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随机图的二部拉姆齐循环数

对于图G和H，令\(G\xrightarrow {k}H\)表示G 的任何k边着色都包含一个单色H作为子图。令\(\mathcal{G}(K_2(N),p)\)是边概率为p 的随机图空间，其中\(K_2(N)\)是完整的\(N\times N\)二部图。对于常数\(p\in (0,1]\)和长度为 2 n的循环\(C_{2n}\)，表明\(\Pr [\mathcal{G}(K_2((k +o(1))n),p)\xrightarrow {k} C_{2n}]\rightarrow 1\)为\(n\rightarrow \infty \)，其中\(k=2,3\)。