The generalized Turán number of $ 2 S_\ell $

Abstract

<abstract><p>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 $ kS_\ell $ denote the disjoint union of $ k $ copies of $ S_\ell $. Gan et al. and Chase determined $ ex(n, K_s, S_\ell) $ for all integers $ s\ge 3 $, $ \ell\ge 1 $ and $ n\ge 1 $. In this paper, we determine $ ex(n, K_s, 2S_\ell) $ for all integers $ s\ge 4 $, $ \ell\ge 1 $ and $ n\ge 1 $.</p></abstract>