Density Hajnal--Szemerédi theorem for cliques of size four

Abstract

The celebrated Corrádi--Hajnal Theorem~\cite{CH63} and the Hajnal--Szemerédi Theorem~\cite{HS70} determined the exact minimum degree thresholds for a graph on $n$ vertices to contain $k$ vertex-disjoint copies of $K_r$, for $r=3$ and general $r \ge 4$, respectively. The edge density version of the Corrádi--Hajnal Theorem was established by Allen--Böttcher--Hladký--Piguet~\cite{ABHP15} for large $n$. Remarkably, they determined the four classes of extremal constructions corresponding to different intervals of $k$. They further proposed the natural problem of establishing a density version of the Hajnal--Szemerédi Theorem: For $r \ge 4$, what is the edge density threshold that guarantees a graph on $n$ vertices contains $k$ vertex-disjoint copies of $K_r$ for $k \le n/r$. They also remarked, ``We are not even sure what the complete family of extremal graphs should be.'' We take the first step toward this problem by determining asymptotically the five classes of extremal constructions for $r=4$. Furthermore, we propose a candidate set comprising $r+1$ classes of extremal constructions for general $r \ge 5$.