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Lower Bounds for Small Ramsey Numbers on Hypergraphs

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Book cover Computing and Combinatorics (COCOON 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11653))

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Abstract

The Ramsey number \(r_k(p, q)\) is the smallest integer N that satisfies for every red-blue coloring on k-subsets of [N], there exist p integers such that any k-subset of them is red, or q integers such that any k-subset of them is blue. In this paper, we study the lower bounds for small Ramsey numbers on hypergraphs by constructing counter-examples and recurrence relations. We present a new algorithm to prove lower bounds for \(r_k(k+1, k+1)\). In particular, our algorithm is able to prove \(r_5(6,6) \ge 72\), where there is no lower bound on 5-hypergraphs before this work. We also provide several recurrence relations to calculate lower bounds based on lower bound values on smaller p and q. Combining both of them, we achieve new lower bounds for \(r_k(p, q)\) on arbitrary p, q, and \(k \ge 4\).

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Notes

  1. 1.

    \(\mathbf {v_2} \ge _c \mathbf {v_1}\) reads “\(v_2\) contains \(v_1\)”.

  2. 2.

    Conventionally, \(\mathbf {1}^n\) is a vector of length n with all coordinates being 1; \(\mathbf {e_i}\) is a vector with the i-th coordinate being 1 and others being 0.

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Acknowledgments

The author wants to thank the anonymous reviewers for their valuable comments. Research at Princeton University partially supported by an innovation research grant from Princeton and a gift from Microsoft.

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Authors and Affiliations

  1. Princeton University, Princeton, NJ, 08540, USA

    S. Cliff Liu

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  1. S. Cliff Liu
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Correspondence to S. Cliff Liu .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Ding-Zhu Du

  2. Xidian University, Xi'an, China

    Zhenhua Duan

  3. Xidian University, Xi'an, China

    Cong Tian

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Cliff Liu, S. (2019). Lower Bounds for Small Ramsey Numbers on Hypergraphs. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_34

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  • DOI: https://doi.org/10.1007/978-3-030-26176-4_34

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-26175-7

  • Online ISBN: 978-3-030-26176-4

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