Multicolor Ramsey numbers for triple systems

Maria Axenovich, A. Gyárfás, Hong Liu, Dhruv Mubayi

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

Given an r-uniform hypergraph H, the multicolor Ramsey number rk(H) is the minimum n such that every k-coloring of the edges of the complete r-uniform hypergraph Knr yields a monochromatic copy of H. We investigate rk(H) when k grows and H is fixed. For nontrivial 3-uniform hypergraphs H, the function rk(H) ranges from 6k(1+o(1)) to double exponential in k. We observe that rk(H) is polynomial in k when H is r-partite and at least single-exponential in k otherwise. Erdos, Hajnal and Rado gave bounds for large cliques Ksr with s≥s0(r), showing its correct exponential tower growth. We give a proof for cliques of all sizes, s>r, using a slight modification of the celebrated stepping-up lemma of Erdos and Hajnal. For 3-uniform hypergraphs, we give an infinite family with sub-double-exponential upper bound and show connections between graph and hypergraph Ramsey numbers. Specifically, we prove thatrk( K3)≤r4k(K43-e)≤r4k(K3)+1, where K43-e is obtained from K43 by deleting an edge. We provide some other bounds, including single-exponential bounds for F5={abe,abd,cde} as well as asymptotic or exact values of rk(H) when H is the bow {abc,ade}, kite {abc,abd}, tight path {abc,bcd,cde} or the windmill {abc,bde,cef,bce}. We also determine many new "small" Ramsey numbers and show their relations to designs. For example, the lower bound for r6(kite)=8 is demonstrated by decomposing the triples of a seven element set into six partial STS (two of them are Fano planes).

Original languageEnglish
Pages (from-to)69-77
Number of pages9
JournalDiscrete Mathematics
Volume322
Issue number1
DOIs
Publication statusPublished - May 6 2014

Fingerprint

Triple System
Ramsey number
Uniform Hypergraph
Coloring
Towers
Polynomials
Erdös
Strombus or kite or diamond
Clique
Fano plane
Exponential Bound
H-function
Hypergraph
Colouring
Lemma
Lower bound
Upper bound
Partial
Path
Polynomial

Keywords

  • Multicolor Ramsey
  • Triple system

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Theoretical Computer Science

Cite this

Multicolor Ramsey numbers for triple systems. / Axenovich, Maria; Gyárfás, A.; Liu, Hong; Mubayi, Dhruv.

In: Discrete Mathematics, Vol. 322, No. 1, 06.05.2014, p. 69-77.

Research output: Contribution to journalArticle

Axenovich, Maria ; Gyárfás, A. ; Liu, Hong ; Mubayi, Dhruv. / Multicolor Ramsey numbers for triple systems. In: Discrete Mathematics. 2014 ; Vol. 322, No. 1. pp. 69-77.
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