On the extremal graphs in generalized Tur\'an problems

TL;DR

This paper characterizes extremal graphs that maximize the number of copies of a graph H in large F-free graphs, advancing the understanding of generalized Turán problems in extremal graph theory.

Contribution

It provides explicit extremal graphs for generalized Turán problems for large n, identifying structures that maximize copies of H while avoiding F.

Findings

Explicit extremal graphs constructed for large n

Maximization of copies of H in F-free graphs established

Advances understanding of generalized Turán problems

Abstract

Given two graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in an $n$-vertex $F$-free graph. For every $F$ and sufficiently large $n$, we present an extremal graph for a generalized Tur\'an problem, i.e., an $F$-free $n$ vertex graph $G$ that for some $H$ contains exactly $\mathrm{ex}(n,H,F)$ copies of $H$.