On Tur\'an-type problems and the abstract chromatic number

TL;DR

This paper develops a combinatorial approach to Turán-type problems and the abstract chromatic number, extending existing theorems to new parameters and settings, and providing stability and exact results.

Contribution

It introduces a purely combinatorial method, extends Erdős-Stone-Simonovits-type theorems to various graph parameters, and achieves stability, supersaturation, and exact results.

Findings

Extended theorems to new graph parameters

Provided stability and supersaturation results

Achieved exact asymptotic results in certain cases

Abstract

In 2020, Coregliano and Razborov introduced a general framework to study limits of combinatorial objects, using logic and model theory. They introduced the abstract chromatic number and proved/reproved multiple Erd\H{o}s-Stone-Simonovits-type theorems in different settings. In 2022, Coregliano extended this by showing that similar results hold when we count copies of $K_t$ instead of edges. Our aim is threefold. First, we provide a purely combinatorial approach. Second, we extend their results by showing several other graph parameters and other settings where Erd\H{o}s-Stone-Simonovits-type theorems follow. Third, we go beyond determining asymptotics and obtain corresponding stability, supersaturation, and sometimes even exact results.