Some sharp results on the generalized Turán numbers

Abstract

For graphs T,H, let ex(n,T,H) denote the maximum number of copies of T in an n-vertex H-free graph. In this paper we prove some sharp results on this generalization of Turán numbers, where our focus is for the graphs T,H satisfying χ(T)<χ(H). This can be dated back to Erdős (1962), where he generalized the celebrated Turán’s theorem by showing that for any r≥m, the Turán graph Tr(n) uniquely attains ex(n,Km,Kr+1). For general graphs H with χ(H)=r+1>m, Alon and Shikhelman (2016) showed that ex(n,Km,H)=rm(nr)m+o(nm). Here we determine this error term o(nm) up to a constant factor. We prove that ex(n,Km,H)=rm(nr)m+biex(n,H)⋅Θ(nm−2), where biex(n,H) is the Turán number of the decomposition family of H. As a special case, we extend Erdős’ result, by showing that Tr(n) uniquely attains ex(n,Km,H) for any edge-critical graph H. We also consider T being non-clique, where even the simplest case seems to be intricate. Following from a more general result, we show that for all s≤t, T2(n) maximizes the number of Ks,t in n-vertex triangle-free graphs if and only if t<s+12+2s+14.