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Minimum Degrees and Codegrees of Ramsey-Minimal 3-Uniform Hypergraphs*†

Part of: Extremal combinatorics

Published online by Cambridge University Press:  20 January 2016

DENNIS CLEMENS  and
YURY PERSON
DENNIS CLEMENS
Affiliation:
Technische Universität Hamburg–Harburg, Institut für Mathematik, Am Schwarzenberg-Campus 3, 21073 Hamburg, Germany (e-mail: [email protected])
YURY PERSON
Affiliation:
Goethe-Universität, Institut für Mathematik, Robert-Mayer-Straß e 10, 60325 Frankfurt am Main, Germany (e-mail: [email protected])
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Abstract

A uniform hypergraph H is called k-Ramsey for a hypergraph F if, no matter how one colours the edges of H with k colours, there is always a monochromatic copy of F. We say that H is k-Ramsey-minimal for F if H is k-Ramsey for F but every proper subhypergraph of H is not. Burr, Erdős and Lovasz studied various parameters of Ramsey-minimal graphs. In this paper we initiate the study of minimum degrees and codegrees of Ramsey-minimal 3-uniform hypergraphs. We show that the smallest minimum vertex degree over all k-Ramsey-minimal 3-uniform hypergraphs for Kt(3) is exponential in some polynomial in k and t. We also study the smallest possible minimum codegree over 2-Ramsey-minimal 3-uniform hypergraphs.

MSC classification

Primary: 05D10: Ramsey theory
Secondary: 05D40: Probabilistic methods
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 6 , November 2016 , pp. 850 - 869
Copyright
Copyright © Cambridge University Press 2016 

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Footnotes

*

After this paper was accepted and processed, we managed to obtain BEL-gadgets for uniformities r ⩾ 4. This work will appear elsewhere.

†

An extended abstract of this paper appears in the proceedings of EuroComb 2015 [3].

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