Embedding clique-factors in graphs with low ℓ-independence number

Abstract

The following question was proposed by Nenadov and Pehova and reiterated by Knierim and Su: given μ>0 and integers ℓ,r and n with n∈rN, is it true that there exists an α>0 such that every n-vertex graph G with δ(G)≥max⁡{12,r−ℓr}n+μn and αℓ(G)≤αn contains a Kr-factor? We give a negative answer to this question for the case ℓ≥3r4 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,ℓ with r>ℓ≥34r and μ>0, there exist α>0 and N such that for every n∈rN with n>N, every n-vertex graph G with δ(G)≥(12−ϱℓ(r−1)+μ)n and αℓ(G)≤αn contains a Kr-factor. Here ϱℓ(r−1) is the Ramsey–Turán density for Kr−1 under the ℓ-independence number condition.