完全子图的重数轮廓
On the Profile of Multiplicities of Complete Subgraphs
摘要：
设$G$是一个完全图在$n$顶点上的$2$ -染色，对于足够大的$n$ .我们证明了$G$包含至少$n^{(\frac{1}{4} - o(1))\log n}$单色完全子图的大小$r$，其中\[ 0.3\log n < r < 0.7\log n. \]以前已知的下限的单色完全子图的总数，由于Székely是$n^{0.1576\log n}$。我们还证明了$G$至少包含$n^{\frac{1}{7} \log n} $大小为$\frac{1}{2}\log n$的单色完全子图。如果进一步假设$G$中最大的单色完全子图的大小为$(\frac{1}{2} + o(1))\log n$（这是一个众所周知的公开问题，这样的图是否存在），那么对于每个常数$0 \le c \le \frac{1}{2}$，我们确定（直到低阶项）大小为$c \log n$的单色完全子图的数量。我们这样做是通过证明一个下界相匹配（低序项）以前的上限塞凯伊。例如，大小为$\frac{1}{2} \log n$的单色完全子图的个数为$n^{\frac{1}{8}(4 - \log e \pm o(1))\log n} \simeq n^{0.32 \log n}$。
英文摘要：
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 of size $r$, where \[ 0.3\log n < r < 0.7\log n. \] The previously known lower bound on the total number of monochromatic complete subgraphs, due to Székely was $n^{0.1576\log n}$. We also prove that $G$ contains at least $n^{\frac{1}{7} \log n} $ monochromatic complete subgraphs of size $\frac{1}{2}\log n$. 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 $n^{\frac{1}{8}(4 - \log e \pm o(1))\log n} \simeq n^{0.32 \log n}$.
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