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A Survey of Hypergraph Ramsey Problems

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Discrete Mathematics and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 165))

Abstract

The classical hypergraph Ramsey number r k(s, n) is the minimum N such that for every red-blue coloring of the k-tuples of {1, …, N}, there are s integers such that every k-tuple among them is red, or n integers such that every k-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of r k(s, n). Our focus is on recent developments and open problems.

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Notes

  1. 1.

    The main result in [48], known as the Happy Ending Theorem, states that for any positive integer n, any sufficiently large set of points in the plane in general position has a subset of n members that form the vertices of a convex polygon.

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Acknowledgements

D. Mubayi’s research partially supported by NSF grant DMS-1300138. A. Suk was supported by NSF grant DMS-1800736, an NSF CAREER award, and an Alfred Sloan Fellowship.

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Authors and Affiliations

  1. Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, IL, USA

    Dhruv Mubayi

  2. Department of Mathematics, University of California at San Diego, La Jolla, CA, USA

    Andrew Suk

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  1. Dhruv Mubayi
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  1. Moscow Institute of Physics and Technology, Dolgoprudny, Russia

    Andrei M. Raigorodskii

  2. Institute of Mathematics, University of Zurich, Zurich, Eswatini

    Michael Th. Rassias

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Mubayi, D., Suk, A. (2020). A Survey of Hypergraph Ramsey Problems. In: Raigorodskii, A.M., Rassias, M.T. (eds) Discrete Mathematics and Applications. Springer Optimization and Its Applications, vol 165. Springer, Cham. https://doi.org/10.1007/978-3-030-55857-4_16

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