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On Some Open Questions for Ramsey and Folkman Numbers

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Graph Theory

Part of the book series: Problem Books in Mathematics ((PBM))

Abstract

We discuss some of our favorite open questions about Ramsey numbers and a related problem on edge Folkman numbers. For the classical two-color Ramsey numbers, we first focus on constructive bounds for the difference between consecutive Ramsey numbers. We present the history of progress on the Ramsey number R(5, 5) and discuss the conjecture that it is equal to 43. For the multicolor Ramsey numbers, we focus on the growth of R r (k), in particular for k = 3. Two concrete conjectured cases, R(3, 3, 3, 3) = 51 and R(3, 3, 4) = 30, are discussed in some detail. For Folkman numbers, we present the history, recent developments, and potential future progress on F e (3, 3; 4), defined as the smallest number of vertices in any K 4-free graph which is not a union of two triangle-free graphs. Although several problems discussed in this paper are concerned with concrete cases and some involve significant computational approaches, there are interesting and important theoretical questions behind each of them.

*Supported by the National Natural Science Foundation (11361008) and the Guangxi Natural Science Foundation (2011GXNSFA018142).

†Work done while on sabbatical at the Gdańsk University of Technology, supported by a grant from the Polish National Science Centre grant 2011/02/A/ST6/00201. Support by the Institut Mittag-Leffler, Djursholm, Sweden, is gratefully acknowledged.

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Acknowledgements

We are grateful to the anonymous reviewers whose comments helped us to improve the presentation of this survey.

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Authors and Affiliations

  1. Guangxi Academy of Sciences, Nanning, Guangxi, 530007, China

    Xiaodong Xu

  2. Department of Algorithms and System Modeling, Faculty of Electronics, Telecommunications and Informatics, Gdańsk University of Technology, 80-233, Gdańsk, Poland

    Stanisław P. Radziszowski

  3. Department of Computer Science, Rochester Institute of Technology, Rochester, NY, 14623, USA

    Stanisław P. Radziszowski

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  1. Xiaodong Xu
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Correspondence to Stanisław P. Radziszowski .

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  1. Department of Applied Mathematics, Naval Postgraduate School, Monterey, California, USA

    Ralucca Gera

  2. School of Computing, Clemson University, Clemson, South Carolina, USA

    Stephen Hedetniemi

  3. Department of Mathematics, Virginia Commonwealth University, Richmond, Virginia, USA

    Craig Larson

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Xu, X., Radziszowski, S.P. (2016). On Some Open Questions for Ramsey and Folkman Numbers. In: Gera, R., Hedetniemi, S., Larson, C. (eds) Graph Theory. Problem Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-31940-7_4

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