### 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 that^{rk}( ^{K3})≤r4_{k}(K43-e)≤r4_{k}(^{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 language | English |
---|---|

Pages (from-to) | 69-77 |

Number of pages | 9 |

Journal | Discrete Mathematics |

Volume | 322 |

Issue number | 1 |

DOIs | |

Publication status | Published - May 6 2014 |

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### Keywords

- Multicolor Ramsey
- Triple system

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Discrete Mathematics*,

*322*(1), 69-77. https://doi.org/10.1016/j.disc.2014.01.004

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

Research output: Contribution to journal › Article

*Discrete Mathematics*, vol. 322, no. 1, pp. 69-77. https://doi.org/10.1016/j.disc.2014.01.004

}

TY - JOUR

T1 - Multicolor Ramsey numbers for triple systems

AU - Axenovich, Maria

AU - Gyárfás, A.

AU - Liu, Hong

AU - Mubayi, Dhruv

PY - 2014/5/6

Y1 - 2014/5/6

N2 - 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).

AB - 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).

KW - Multicolor Ramsey

KW - Triple system

UR - http://www.scopus.com/inward/record.url?scp=84893341454&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84893341454&partnerID=8YFLogxK

U2 - 10.1016/j.disc.2014.01.004

DO - 10.1016/j.disc.2014.01.004

M3 - Article

VL - 322

SP - 69

EP - 77

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 1

ER -