Every graph is eventually Turán-good

Abstract

Let H be a graph. We show that if r is large enough as a function of H, then the r-partite Turán graph maximizes the number of copies of H among all K r+1 -free graphs on a given number of vertices. This confirms a conjecture of Gerbner and Palmer.

Generalized Turán problems for double stars

“…We can deal with counting multiple graphs at the same time if the same graph is extremal for each of them, in particular if the same graph is extremal for each subgraphs of H. Two examples of this are counting linear forests if F has a color-critical edge [7], and counting any graph H if F has a color-critical edge and chromatic number at least 300|V (H)| 9 +1 [15,11].…”

A localized approach to generalized Turán problems

“…The quantities ex(n, H, F ) and mex(m, H, F ) have motivated many interesting results; see [5,6,11,12,13,20] for an (incomplete) sample. However, these problems often stretch the notion of "easily measurable" properties on which extremal graph theory is based.…”

Subgraph Densities in $K_r$-Free Graphs

“…Recently Morrison, Nir, Norin, Rzążewski and Wesolek [12] showed that for any graph H and large enough r, the maximum number of copies of H in a K r -free n-vertex graph is obtained by the Turán graph T r−1 (n), the balanced blow-up of K r−1 . In other words, the above conjecture works if r is enough large comparing to χ(H).…”