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arXiv · 2410.09617

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

Dániel Gerbner·Hilal Hama Karim·Gaurav Kucheriya
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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.

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Dániel Gerbner, Hilal Hama Karim, Gaurav Kucheriya. 2024-10-12. On Tur\'an-type problems and the abstract chromatic number. https://arxiv.org/abs/2410.09617

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