eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2024-09-16
59:1
59:18
10.4230/LIPIcs.APPROX/RANDOM.2024.59
article
Ramsey Properties of Randomly Perturbed Hypergraphs
Aigner-Horev, Elad
1
https://orcid.org/0000-0002-9207-0596
Hefetz, Dan
1
https://orcid.org/0000-0001-8923-3879
Schacht, Mathias
2
School of Computer Science, Ariel University, Israel
Fachbereich Mathematik, Universität Hamburg, Germany
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.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol317-approx-random2024/LIPIcs.APPROX-RANDOM.2024.59/LIPIcs.APPROX-RANDOM.2024.59.pdf
Ramsey Theory
Smoothed Analysis
Random Hypergraphs