On the generalized Turán number of star forests

Abstract

The generalized Turán number ex$(n,K_s,H)$ is defined to be the maximum number of copies of a complete graph $K_s$ in any $H$-free graph on $n$ vertices. Let $S_{\ell }$ denote the star on $\ell +1$ vertices, and let $k{S}_{\ell }$ denote the disjoint union of $k$ copies of $S_{\ell }$. Gan et al. (2015) and Chase (2020) determined ex$(n,K_s,S_{\ell })$ for $s\ge 3$, $\ell \ge 1$ and $n\ge 1$. In this paper, we consider to investigate the generalized Turán number of star forests. We determine ex$(n,K_s,S_{{\ell }_1}\cup S_{{\ell }_2})$ for $s\ge 4$, ${\ell }_2\le {\ell }_1\le 2{\ell }_2+1$ and $n\ge 1$, and ex$(n,K_s,S_{{\ell }_1}\cup S_{{\ell }_2})$ for $s\ge 4$, ${\ell }_1\ge 2{\ell }_2+2$ and $n\ge 3{\ell }_1^2$. Those imply ex$(n,K_s,2S_{\ell })$ for $s\ge 4$, $\ell \ge 1$ and $n\ge 1$. Moreover, we also determine ex$(n,K_s,3S_{\ell })$ for $s\ge 5$, $\ell \ge 1$ and $n\ge 1$.