Embedding clique-factors in graphs with low $\ell$-independence number

Abstract: The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: Given integers $\ell,r$ and $n$ with $n\in r\mathbb{N}$, is it true that every $n$-vertex graph $G$ with $\delta(G) \ge \max { \frac{1}{2},\frac{r - \ell}{r} }n + o(n)$ and $\alpha_{\ell}(G) = o(n) $ contains a $K_{r}$-factor? We give a negative answer for the case when $\ell\ge \frac{3r}{4}$ by giving a family of constructions using the so-called cover thresholds and show that the minimum degree condition given by our construction is asymptotically best possible. That is, for all integers $r,\ell$ with $r > \ell \ge \frac{3}{4}r$ and $\mu >0$, there exist $\alpha > 0$ and $N$ such that for every $n\in r\mathbb{N}$ with $n>N$, every $n$-vertex graph $G$ with $\delta(G) \ge \left( \frac{1}{2-\varrho_{\ell}(r-1)} + \mu \right)n $ and $\alpha_{\ell}(G) \le \alpha n$ contains a $K_{r}$-factor. Here $\varrho_{\ell}(r-1)$ is the Ramsey--Tur\'an density for $K_{r-1}$ under the $\ell$-independence number condition.