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Beyond Chromatic Threshold via (p,q)-Theorem, and Blow-Up Phenomenon

Authors Hong Liu, Chong Shangguan , Jozef Skokan, Zixiang Xu

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Author Details

Hong Liu
  • Extremal Combinatorics and Probability Group, Institute for Basic Science, Daejeon, South Korea
Chong Shangguan
  • Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, China
  • Frontiers Science Center for Nonlinear Expectations, Ministry of Education, Qingdao, China
Jozef Skokan
  • Department of Mathematics, London School of Economics, UK
Zixiang Xu
  • Extremal Combinatorics and Probability Group, Institute for Basic Science, Daejeon, South Korea

Funding

  • Liu, Hong: Supported by IBS-R029-C4.
  • Shangguan, Chong: Supported by the National Natural Science Foundation of China under Grant Nos. 12101364 and 12231014, and the Natural Science Foundation of Shandong Province under Grant No. ZR2021QA005.
  • Xu, Zixiang: Supported by IBS-R029-C4.

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Hong Liu, Chong Shangguan, Jozef Skokan, and Zixiang Xu. Beyond Chromatic Threshold via (p,q)-Theorem, and Blow-Up Phenomenon. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 71:1-71:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.71

Abstract

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 K_r-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 K_r-free graph G satisfies that the common neighborhood of every pair of non-adjacent vertices induces a subgraph with K_{r-2}-density at least ε > 0, then G must be a blow-up of some K_r-free graph F on at most 2^O(r/ε log1/ε) vertices. Furthermore, this single exponential bound is optimal. We construct examples with no K_r-free homomorphic image of size smaller than 2^Ω_r(1/ε).

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Hypergraphs
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Extremal graph theory
Keywords
  • (p,q)-theorem
  • fractional Helly number
  • weak ε-net
  • chromatic threshold
  • VC dimension

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