Clique-factors in graphs with low $K_{\ell }$-independence number

Abstract

Given $r\in \mathbb{N}$ with $r\ge 4$, we show that there exists $n_0\in \mathbb{N}$ such that for every $n\ge n_0$, every $n$-vertex graph $G$ with $\delta (G)\ge (\frac{1}{2}+o(1))n$ and ${\alpha }_{r-2}(G)=o(n)$ contains a $K_r$-factor. This resolves the first open case of a question proposed by Nenadov and Pehova, and reiterated by Knierm and Su. We further introduce two lower bound constructions that, along with some known results, fully resolve a question presented by Balogh, Molla, and Sharifzadeh.