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