Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-23T23:36:18.389Z Has data issue: false hasContentIssue false
  • English
  • Français

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])
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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].

References

[1] Burr, S. A., Erdős, P. and Lovász, L. (1976) On graphs of Ramsey type. Ars Combin. 1 167–190.Google Scholar
[2] Burr, S. A., Nešetřil, J. and Rödl, V. (1985) On the use of senders in generalized Ramsey theory for graphs. Discrete Math. 54 1–13.Google Scholar
[3] Clemens, D. and Person, Y. (2015) Minimum degrees and codegrees of minimal Ramsey 3-uniform hypergraphs. In Eighth European Conference on Combinatorics, Graph Theory and Applications (EuroComb 2015), Electron. Notes in Discrete Math., Elsevier, pp. 23–30.Google Scholar
[4] Conlon, D. (2009) A new upper bound for diagonal Ramsey numbers. Ann. of Math. (2) 170 941–960.Google Scholar
[5] Conlon, D., Fox, J. and Sudakov, B. (2015) Recent developments in graph Ramsey theory. In Surveys in Combinatorics 2015, Cambridge University Press, pp. 49–118.CrossRefGoogle Scholar
[6] Dudek, A., La Fleur, S., Mubayi, D. and Rödl, V. On the size-Ramsey number of hypergraphs. Submitted.Google Scholar
[7] Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1978) The size Ramsey number. Period. Math. Hungar. 9 145–161.CrossRefGoogle Scholar
[8] Erdős, P. and Hajnal, A. (1966) On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. 17 61–99.Google Scholar
[9] Fox, J., Grinshpun, A., Liebenau, A., Person, Y. and Szabó, T. (2014) What is Ramsey-equivalent to a clique? J. Combin. Theory Ser. B 109 120–133.Google Scholar
[10] Fox, J., Grinshpun, A., Liebenau, A., Person, Y. and Szabó, T. On the minimum degree of minimal Ramsey graphs for multiple colours. J. Combin. Theory Ser. B, accepted.Google Scholar
[11] Fox, J. and Lin, K. (2006) The minimum degree of Ramsey-minimal graphs. J. Graph Theory 54 167–177.Google Scholar
[12] Graham, R. L., Rothschild, B. L. and Spencer, J. H. (1990) Ramsey Theory, second edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.Google Scholar
[13] Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley.CrossRefGoogle Scholar
[14] Ramsey, F. P. (1930) On a problem in formal logic. Proc. Lond. Math. Soc. 30 264–286.Google Scholar
[15] Rödl, V. and Siggers, M. (2008) On Ramsey minimal graphs. SIAM J. Discrete Math. 22 467–488.Google Scholar
[16] Spencer, J. (1975) Ramsey's theorem: A new lower bound. J. Combin. Theory Ser. A 18 108–115.Google Scholar
[17] Szabó, T., Zumstein, P. and Zürcher, S. (2010) On the minimum degree of minimal Ramsey graphs. J. Graph Theory 64 150–164.Google Scholar