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Beyond chromatic threshold via (p,q)-theorem, and blow-up phenomenon

Liu, Hong, Shangguan, Chong, Skokan, Jozef ORCID: 0000-0003-3996-7676 and Xu, Zixiang (2024) Beyond chromatic threshold via (p,q)-theorem, and blow-up phenomenon. In: Mulzer, Wolfgang and Phillips, Jeff M., (eds.) 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (293). Schloss Dagstuhl - Leibniz-Zentrum für Informatik. (In Press)

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We establish a novel connection between the well-known chromatic threshold problem in extremal combinatorics and the celebrated (p,q)-theorem in discrete geometry. In particular, for a graph G with bounded clique number and a natural density condition, we prove a (p,q)-theorem for an abstract convexity space associated with G. Our result strengthens those of Thomassen and Nikiforov on the chromatic threshold of cliques. Our (p,q)-theorem can also be viewed as a χ-boundedness result for (what we call) ultra maximal Kr-free graphs. We further show that the graphs under study are blow-ups of constant size graphs, improving a result of Oberkampf and Schacht on homomorphism threshold of cliques. Our result unravels the cause underpinning such a blow-up phenomenon, differentiating the chromatic and homomorphism threshold problems for cliques. Our result implies that for the homomorphism threshold problem, rather than the minimum degree condition usually considered in the literature, the decisive factor is a clique density condition on co-neighborhoods of vertices. More precisely, we show that if an n-vertex Kr-free graph G satisfies that the common neighborhood of every pair of non-adjacent vertices induces a subgraph with Kr-2-density at least ε > 0, then G must be a blow-up of some Kr-free graph F on at most 2O (r/ε log 1/ε) vertices. Furthermore, this single exponential bound is optimal. We construct examples with no Kr-free homomorphic image of size smaller than 2Ωr (1/ε) .

Item Type: Book Section
Additional Information: © 2024 The Authors
Divisions: Mathematics
Subjects: Q Science > QA Mathematics
Date Deposited: 16 Apr 2024 09:18
Last Modified: 16 May 2024 06:02

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