Clique-factors in graphs with sublinear $\ell$-independence number

Abstract

Given a graph $G$ and an integer $\ell \ge 2$, we denote by $\hat{I}{\pm }_{\ell }(G)$ the maximum size of a $K_{\ell }$-free subset of vertices in $V(G)$. A recent question of Nenadov and Pehova asks for determining the best possible minimum degree conditions forcing clique-factors in $n$-vertex graphs $G$ with $\hat{I}{\pm }_{\ell }(G)=o(n)$, which can be seen as a Ramsey--Turán variant of the celebrated Hajnal--Szemerédi theorem. In this paper we find the asymptotical sharp minimum degree threshold for $K_r$-factors in $n$-vertex graphs $G$ with $\hat{I}{\pm }_{\ell }(G)=n^{1-o(1)}$ for all $r\ge \ell \ge 2$.

arXiv admin note: text overlap with arXiv:2111.10512