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


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.

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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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This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[2] S. A. Burr , J. Nešetřil and V. Rödl (1985) On the use of senders in generalized Ramsey theory for graphs. Discrete Math. 54 113.

[4] D. Conlon (2009) A new upper bound for diagonal Ramsey numbers. Ann. of Math. (2) 170 941960.

[5] D. Conlon , J. Fox and B. Sudakov (2015) Recent developments in graph Ramsey theory. In Surveys in Combinatorics 2015, Cambridge University Press, pp. 49118.

[7] P. Erdős , R. J. Faudree , C. C. Rousseau and R. H. Schelp (1978) The size Ramsey number. Period. Math. Hungar. 9 145161.

[8] P. Erdős and A. Hajnal (1966) On chromatic number of graphs and set-systems. Acta Math. Acad. Sci. Hungar. 17 6199.

[9] J. Fox , A. Grinshpun , A. Liebenau , Y. Person and T. Szabó (2014) What is Ramsey-equivalent to a clique? J. Combin. Theory Ser. B 109 120133.

[11] J. Fox and K. Lin (2006) The minimum degree of Ramsey-minimal graphs. J. Graph Theory 54 167177.

[12] R. L. Graham , B. L. Rothschild and J. H. Spencer (1990) Ramsey Theory, second edition, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley.

[13] S. Janson , T. Łuczak and A. Ruciński (2000) Random Graphs, Wiley.

[15] V. Rödl and M. Siggers (2008) On Ramsey minimal graphs. SIAM J. Discrete Math. 22 467488.

[16] J. Spencer (1975) Ramsey's theorem: A new lower bound. J. Combin. Theory Ser. A 18 108115.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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