Packing edge disjoint cliques in graphs

Abstract

Let $r \ge 3$ be fixed and $G$ be an $n$-vertex graph. A long-standing conjecture of Gy\H{o}ri states that if $e(G) = t_{r-1}(n) + k$, where $t_{r-1}(n)$ denotes the number of edges of the Tur\'{a}n graph on $n$ vertices and $r - 1$ parts, then $G$ has at least $(2 - o(1))k/r$ edge disjoint $r$-cliques. We prove this conjecture.