On Turán-type problems and the abstract chromatic number

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 Erdős-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 Erdős-Stone-Simonovits-type theorems follow. Third, we go beyond determining asymptotics and obtain corresponding stability, supersaturation, and sometimes even exact results.

Theorems & Definitions (27)

• Definition 1.1
• Lemma 1.2
• proof
• Definition 1.3
• Definition 1.4
• Theorem 1.1
• proof
• Theorem 2.1
• proof
• Theorem 2.2