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arXiv 0803.2375math.CO
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非均匀分布

Unavoidable patterns

Jacob Fox, Benny Sudakov

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10 pages, corrected typos
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摘要：

设\mathcal{F}_k表示2k个顶点上的2-边着色完全图族，其中一种颜色形成一个k阶团或两个k阶不相交团. Bollobás证明了对任意ε>0和正整数k，存在n（k，ε）使得n阶完全图\geq n（k，ε）的任意2-边染色包含\mathcal{F}_k的一个成员，其中n阶完全图\geq n（k，ε）的每种颜色至少有ε{nPDMATHABMO 2}条边.卡特勒和蒙塔格证明了这个猜想，他们证明了n（k，ε）<4^{k/ε}。我们给出了这个猜想的一个简单得多的证明，另外证明了对于某个常数c，n（k，ε）<ε^{-ck}。对于所有的k和ε，这个界紧到指数中的常数因子。我们也讨论了类似的结果，比赛和超图。

英文摘要：

Let \mathcal{F}_k denote the family of 2-edge-colored complete graphs on 2k vertices in which one color forms either a clique of order k or two disjoint cliques of order k. Bollobás conjectured that for every ε>0 and positive integer k there is an n(k,ε) such that every 2-edge-coloring of the complete graph of order n \geq n(k,ε) which has at least ε{n \choose 2} edges in each color contains a member of \mathcal{F}_k. This conjecture was proved by Cutler and Montágh, who showed that n(k,ε)<4^{k/ε}. We give a much simpler proof of this conjecture which in addition shows that n(k,ε)<ε^{-ck} for some constant c. This bound is tight up to the constant factor in the exponent for all k and ε. We also discuss similar results for tournaments and hypergraphs.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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