SIAM Journal on Discrete Mathematics, Volume 34, Issue 2, Page 1001-1010, January 2020.
A seminal result of Hajnal and Szemerédi states that if a graph $G$ with $n$ vertices has minimum degree $\delta(G) \ge (r-1)n/r$ for some integer $r \ge 2$, then $G$ contains a $K_r$-factor, assuming $r$ divides $n$. Extremal examples which show optimality of the bound on $\delta(G)$ are very structured and, in particular, contain large independent sets. In analogy to the Ramsey--Turán theory, Balogh, Molla, and Sharifzadeh initiated the study of how the absence of such large independent sets influences sufficient minimum degree. We show the following two related results: (a) For any $r > \ell \ge 2$, if $G$ is a graph satisfying $\delta(G) \ge \frac{r - \ell}{r - \ell + 1}n + \Omega(n)$ and $\alpha_\ell(G) =o(n)$, that is, a largest $K_\ell$-free induced subgraph has at most $o(n)$ vertices, then $G$ contains a $K_r$-factor. This is optimal for $\ell = r - 1$ and extends a result of Balogh, Molla, and Sharifzadeh who considered the case $r = 3$. (b) If a graph $G$ satisfies $\delta(G) = \Omega(n)$ and $\alpha_r^*(G) =o(n)$, that is, every induced $K_r$-free $r$-partite subgraph of $G$ has at least one vertex class of size $o(n)$, then it contains a $K_r$-factor. A similar statement is proven for a general graph $H$.

中文翻译：

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Hajnal-Szemerédi定理的Ramsey-Turán变体

SIAM离散数学杂志，第34卷，第2期，第1001-1010页，2020年1月。
Hajnal和Szemerédi的开创性结果指出，如果具有$ n $个顶点的图$ G $的最小度为某个整数$ r \ ge 2 $，则其度数为\\ delta（G）\ ge（r-1）n / r $，则假设$ r $除以$ n $，则$ G $包含$ K_r $因子。显示$ \ delta（G）$上界的最优性的极端示例非常结构化，特别是包含大的独立集。与Ramsey-Turán理论类似，Balogh，Molla和Sharifzadeh发起了一项研究，研究缺乏如此大的独立集合如何影响足够的最低程度。我们显示以下两个相关结果：（a）对于任何$ r> \ ell \ ge 2 $，如果$ G $是满足$ \ delta（G）\ ge \ frac {r-\ ell} {r- \ ell + 1} n + \ Omega（n）$和$ \ alpha_ \ ell（G）= o（n）$，也就是说，最大的无K_ell诱导子图最多具有$ o（n ）$个顶点，则$ G $包含一个$ K_r $因子。这对于$ \ ell = r-1 $是最佳的，并且扩展了考虑了$ r = 3 $的情况的Balogh，Molla和Sharifzadeh的结果。（b）如果图$ G $满足$ \ delta（G）= \ Omega（n）$和$ \ alpha_r ^ *（G）= o（n）$，即每个诱导的$ K_r $ -free $ $ G $的r $ -partite子图至少具有一个大小为$ o（n）$的顶点类，然后它包含$ K_r $-因子。对于一般图形$ H $也证明了类似的陈述。