发表机构
University of Warwick; Nanjing University; Nanjing University of Aeronautics and Astronautics（华威大学; 南京大学; 南京航空航天大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究超图扩张的Turán密度，确定了H_k^r的Turán密度的常数因子，证明k/log r→∞时π(H_k^r)=(1+o(1))k/r，还得到3-色图F的π(F(r))下界接近上界1/r。
AI 中文摘要
对于t-图F和整数r≥t，\u201cr-扩张\u201dF(r)是通过向F的每条边添加相同的r-t个新顶点得到的r-图。我们研究Turán密度π(F(r))的行为，其中π(F(r))是不含F(r)的大型r-图的渐近最大边密度，将其作为r的函数。令H_k^r为k个顶点的完全(k-1)-图的r-扩张；等价地，H_k^r是（同构意义下）有r+1个顶点和k条边的r-图。本文确定了每个r-图H_k^r的Turán密度的常数因子，并估计了来自Sidorenko环形构造的π(H_k^r)的下界。特别地，证明了当k/log r→∞时，π(H_k^r)=(1+o(1))k/r。此外，我们考虑图的扩张（即t=2的情况），并证明对于每个3-色图F，π(F(r))≥(log r)^{-1-o(1)}/r，这接近已知的上界1/r。注意，对于所有其他图F，π(F(r))的函数要么恒为0，要么一致有界且远离0。
英文摘要
For a $t$-graph $F$ and an integer $r\ge t$, the \emph{$r$-augmentation} $F(r)$ is the $r$-graph obtained by adding the same set of $r-t$ new vertices to every edge of $F$. We investigate the behaviour of the \emph{Turán density $π(F(r))$}, which is the asymptotically maximum edge density of a large $F(r)$-free $r$-graph, as a function of $r$. Let $H_k^r$ be the $r$-augmentation of the complete $(k-1)$-graph on $k$ vertices; equivalently, $H_k^r$ is the (unique up to isomorphism) $r$-graph with $r+1$ vertices and $k$ edges. This paper determines the Turán density of each $r$-graph $H_k^r$ within a constant factor and estimates the lower bounds on $π(H_k^r)$ coming from the circular construction of Sidorenko. In particular, it is shown that if $k/\log r\to\infty$ then $π(H_k^r)=(1+o(1))k/r$. Also, we consider augmentations of graphs (that is, the case $t=2$) and prove that $π(F(r))\ge(\log r)^{-1-o(1)}/r$ for every 3-chromatic graph $F$, which comes close to the known upper bound $1/r$. Note that, for all other graphs $F$, the function $π(F(r))$ is known to be either identically 0 or uniformly bounded away from 0.
