*东雷 杨 (山东大学)
Given an integer $r\ge 2$ and a graph $G$, let $\alpha_{r}(G)$ be the maximum size of a $K_{r}$-free subset of vertices and write $\alpha(G)=\alpha_2(G)$. A central topic in Ramsey--Tur\'{a}n theory, initiated by Erd\H{o}s and S\'{o}s, is to determine $RT_{r}(n;H; o(n))$, the minimum number of edges which guarantees that every $n$-vertex graph $G$ with $\alpha_{r}(G) = o(n)$ contains a copy of $H$. For a $k$-vertex graph $F$ and a graph $G$, an $F$-tiling is a collection of vertex-disjoint copies of $F$ in $G$. We call an $F$-tiling perfect if it covers the vertex set of $G$. We will also refer to a perfect $F$-tiling as an $F$-factor, which is a fundamental object in graph theory with a wealth of results from various aspects. Motivated by Ramsey-Turán theory, a recent trend has focused on reducing the minimum degree condition forcing the existence of $F$-factors in graphs by adding an extra condition that provides pseudorandom properties. In this talk, we mainly investigate the effect of imposing the condition that $\alpha_{r}(G)=o(n)$ by studying the minimum degree thresholds for $K_k$-tilings, and more generally, $F$-tilings. Similar questions for $F$-factors are considered where the condition $\alpha_{r}(G)=o(n)$ is replaced by $\alpha_{r}^*(G)=o(n)$ ( $r$-partite hole number).
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