AI 中文总结
本文研究Reiher团密度定理的局部版本，证明无H图中的局部团密度定理，给出无H图中顶点子集含K_t副本的精确下界，该界在无穷多n值下可达到。
AI 中文摘要
2016年，Reiher的团密度定理确定了具有指定边密度的图中K_t副本的最小数量。本文研究其局部版本，证明了无H图中的局部团密度定理如下：对于满足2≤t≤r-1的整数r和t、任意r色图H、满足(t-2)/(2(t-1))≤γ≤(r-2)/(2(r-1))的实数γ以及0≤α≤1的实数α，确定最大值β:=β(r,t,α,γ)，使得对于每个具有至少γn²条边的n顶点无H图G，G中每个⌈αn⌉顶点子集包含至少(β-o(1))n^t个K_t副本。特别地，当H=K_r时，每个⌈αn⌉顶点子集包含至少⌊βn^t⌋个K_t副本，这是一个精确界。对于α和γ的合适选择，即对应极值构造中的所有部分比例均为有理数时，该界在无穷多个n值下可达到。
英文摘要
In 2016, Reiher's clique density theorem determined the minimum number of copies of $K_t$ in a graph with a prescribed edge density. In this paper, we investigate its local version and prove a local clique density theorem in $H$-free graphs as follows. For integers $r$ and $t$ with $2\leq t\leq r-1$, any $r$-chromatic graph $H$, any real numbers $γ$ and $α$ with $\frac{t-2}{2(t-1)}\leqγ\leq \frac{r-2}{2(r-1)}$ and $0\leqα\leq 1$, we determine the maximum value $β:=β(r,t,α,γ)$ such that for every $n$-vertex $H$-free graph $G$ with at least $γn^2$ edges, every $\lceilαn\rceil$-vertex subset in $G$ contains at least $(β-o(1))n^{t}$ copies of $K_t$. In particular, when $H=K_r$, every $\lceilαn\rceil$-vertex subset contains at least $\lfloorβn^t\rfloor$ copies of $K_t$, which is an exact bound. For suitable choices of $α$ and $γ$, namely, those for which all part ratios in the corresponding extremal construction are rational, this bound is attained for infinitely many values of $n$.
