Unified approach to the generalized Tur\'an problem and supersaturation

TL;DR

This paper introduces a unifying framework for the generalized Turán problem and supersaturation in graph theory, exploring extremal functions, proof methods, and phase transitions, with connections to hypergraph extremal questions.

Contribution

It provides a comprehensive survey, new results, and insights into proof techniques and extremal structures in the generalized Turán and supersaturation problems.

Findings

New bounds for supersaturation-extremal functions

Identification of phase transition phenomena in extremal structures

Connections established with hypergraph extremal problems

Abstract

In this paper we introduce a unifying approach to the generalized Tur\'an problem and supersaturation results in graph theory. The supersaturation-extremal function $satex(n, F : m, G)$ is the least number of copies of a subgraph $G$ an $n$-vertex graph can have, which contains at least $m$ copies of $F$ as a subgraph. We present a survey, discuss previously known results and obtain several new ones focusing mainly on proof methods, extremal structure and phase transition phenomena. Finally we point out some relation with extremal questions concerning hypergraphs, particularly Berge-type results.