The classic extremal problem is that of computing the maximum number of edges in an F -free graph. In the case where F = K r+1 , the extremal number was determined by Turán. Later results, known as supersaturation theorems, proved that in a graph containing more edges than the extremal number, there must also be many copies of K r+1 . Alon and Shikhelman introduced a broader class of problems asking for the maximum number of copies of a graph T in an F -free graph. In this paper, we determine some of these generalized extremal numbers and prove supersaturation results for them.

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

No 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 ask for the maximum number of copies of a graph H in an n-vertex, F -free graph, denoted by ex(n, H, F ). We show how to extend the new, localized approach of Bradač, Malec, and Tompkins to generalized Turán problems. We weight the copies of H (typically taking H = K t ), instead of the edges, based on the size of the largest clique, path, or star containing the vertices of the copy of H, and in each case prove a tight upper bound on the sum of the weights. A consequence of our new localized theorems is an asymptotic determination of ex(n, H, K 1,r ) for every H having at least one dominating vertex and mex(m, H, K 1,r ) for every H having at least two dominating vertices.