Multicolor Ramsey numbers for triple systems
Maria Axenovich, Andras Gyarfas, Hong Liu, Dhruv Mubayi

TL;DR
This paper explores multicolor Ramsey numbers for 3-uniform hypergraphs, establishing bounds, connections to graph Ramsey numbers, and analyzing specific small hypergraph configurations to advance understanding of hypergraph Ramsey theory.
Contribution
It provides new bounds and exact values for multicolor Ramsey numbers of hypergraphs, extends known results to all clique sizes, and links hypergraph Ramsey numbers to combinatorial designs.
Findings
Established bounds for multicolor Ramsey numbers of hypergraphs.
Connected hypergraph Ramsey numbers to graph Ramsey numbers and designs.
Determined exact and asymptotic values for specific small hypergraphs.
Abstract
Given an -uniform hypergraph , the multicolor Ramsey number is the minimum such that every -coloring of the edges of the complete -uniform hypergraph yields a monochromatic copy of . We investigate when grows and is fixed. For nontrivial 3-uniform hypergraphs , the function ranges from to double exponential in . We observe that is polynomial in when is -partite and at least single-exponential in otherwise. Erd\H{o}s, Hajnal and Rado gave bounds for large cliques with , showing its correct exponential tower growth. We give a proof for cliques of all sizes, , using a slight modification of the celebrated stepping-up lemma of Erd\H{o}s and Hajnal. For 3-uniform hypergraphs, we give an infinite family with sub-double-exponential upper bound and show…
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Topology and Set Theory · Advanced Graph Theory Research
