SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 950-971, January 2020.
Let $G$ be a 2-coloring of a complete graph on $n$ vertices, for sufficiently large $n$. We prove that $G$ contains at least $n^{(\frac{1}{4} - o(1))\log n}$ monochromatic complete subgraphs, thus improving over a lower bound of $n^{0.1576\log n}$ due to Székely [Combinatorica, 4 (1984), pp. 363--372]. We also present lower bounds concerning the number of monochromatic complete subgraphs within certain ranges of sizes, incomparable in nature to lower bounds previously proved by Conlon [Combinatorica, 32 (2012), pp. 171--186]. If furthermore one assumes that the largest monochromatic complete subgraph in $G$ is of size $(\frac{1}{2} + o(1))\log n$ (it is a well known open question whether such graphs exist), then for every constant $0 \le c \le \frac{1}{2}$ we determine (up to low order terms) the number of monochromatic complete subgraphs of size $c \log n$. We do so by proving a lower bound that matches (up to low order terms) a previous upper bound of Székely. For example, the number of monochromatic complete subgraphs of size $\frac{1}{2} \log n$ is $n^{\frac{1}{8}(4 - \log e \pm o(1))\log n} \simeq n^{0.32 \log n}$.

中文翻译：

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关于完整子图的多重性的轮廓

SIAM离散数学杂志，第34卷，第1期，第950-971页，2020年1月。
假设$ G $是$ n $顶点上完整图的2色，对于足够大的$ n $。我们证明$ G $至少包含$ n ^ {（\ frac {1} {4}-o（1））\ log n} $个单色完整子图，从而在$ n ^ {0.1576 \ log n} $归因于Székely[Combinatorica，4（1984），第363--372页]。我们还提出了关于一定尺寸范围内的单色完整子图的数量的下界，这在本质上与先前由Conlon证明的下界无可比拟[Combinatorica，32（2012），pp。171--186]。如果进一步假设$ G $中最大的单色完整子图的大小为$（\ frac {1} {2} + o（1））\ log n $（是否存在这样的图是一个众所周知的公开问题） ，那么对于每个常数$ 0 \ le c \ le \ frac {1} {2} $，我们确定（直至低阶项）大小为$ c \ log n $的单色完整子图的数量。我们通过证明一个下限与Székely的上一个上限匹配（最多低阶项）来实现。例如，大小为$ \ frac {1} {2} \ log n $的单色完整子图的数量为$ n ^ {\ frac {1} {8}（4-\ log e \ pm o（1）） \ log n} \ simeq n ^ {0.32 \ log n} $。