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Ramsey Properties of Randomly Perturbed Hypergraphs

Authors Elad Aigner-Horev , Dan Hefetz , Mathias Schacht

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  • Mathematics of computing → Random graphs
  • Mathematics of computing → Hypergraphs
Keywords
  • Ramsey Theory
  • Smoothed Analysis
  • Random Hypergraphs

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Abstract

We study Ramsey properties of randomly perturbed 3-uniform hypergraphs. For t ≥ 2, write K^(3)_t to denote the 3-uniform expanded clique hypergraph obtained from the complete graph K_t by expanding each of the edges of the latter with a new additional vertex. For an even integer t ≥ 4, let M denote the asymmetric maximal density of the pair (K^(3)_t, K^(3)_{t/2}). We prove that adding a set F of random hyperedges satisfying |F| ≫ n^{3-1/M} to a given n-vertex 3-uniform hypergraph H with non-vanishing edge density asymptotically almost surely results in a perturbed hypergraph enjoying the Ramsey property for K^(3)_t and two colours. We conjecture that this result is asymptotically best possible with respect to the size of F whenever t ≥ 6 is even. The key tools of our proof are a new variant of the hypergraph regularity lemma accompanied with a tuple lemma providing appropriate control over joint link graphs. Our variant combines the so called strong and the weak hypergraph regularity lemmata.

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Elad Aigner-Horev, Dan Hefetz, and Mathias Schacht. Ramsey Properties of Randomly Perturbed Hypergraphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 59:1-59:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.59

Author Details

Elad Aigner-Horev
  • School of Computer Science, Ariel University, Israel
Dan Hefetz
  • School of Computer Science, Ariel University, Israel
Mathias Schacht
  • Fachbereich Mathematik, Universität Hamburg, Germany

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