Turán number of three disjoint cliques of arbitrary sizes

Abstract

The Turán number $\mathrm{ex}(\mathrm{n},\mathrm{H})$ of a given graph $H$ is the maximum number of edges in an $n$-vertex $H$-free graph. In this paper, for all $p\ge q\ge r\ge 2$ and $n\ge p+q+r$, we determine the exact value of $\mathrm{ex}(\mathrm{n},{\mathrm{K}}_{\mathrm{p}}\cup {\mathrm{K}}_{\mathrm{q}}\cup {\mathrm{K}}_{\mathrm{r}})$ and describe all extremal graphs.

arXiv comment26 pages