On supersaturation and stability for generalized Tur\'an problems

TL;DR

This paper introduces a new supersaturation result and a stability theorem for the maximum number of copies of a graph H in an n-vertex F-free graph, focusing on cases where the chromatic number of H is less than that of F.

Contribution

It provides a novel general supersaturation result for $ ext{ex}(n,H,F)$ and a new proof of a stability theorem for $ ext{ex}(n,K_r,F)$ in the context of generalized Turán problems.

Findings

New supersaturation result for $ ext{ex}(n,H,F)$ when $ ext{chi}(H) < ext{chi}(F)$.

A new proof of a stability theorem for $ ext{ex}(n,K_r,F)$.

Enhanced understanding of extremal configurations in generalized Turán problems.

Abstract

Fix graphs $F$ and $H$. Let $\mathrm{ex}(n,H,F)$ denote the maximum number of copies of a graph $H$ in an $n$-vertex $F$-free graph. In this note we will give a new general supersaturation result for $\mathrm{ex}(n,H,F)$ in the case when $\chi(H) < \chi(F)$ as well as a new proof of a stability theorem for $\mathrm{ex}(n,K_r,F)$.