### Abstract

Eroh and Oellermann defined BRR (G_{1}, G_{2}) as the smallest N such that any edge coloring of the complete bipartite graph K_{N,N} contains either a monochromatic G_{1} or a multicolored G_{2}. We restate the problem of determining BRR (K_{1,λ}, K_{r,s}) in matrix form and prove estimates and exact values for several choices of the parameters. Our general bound uses Füredi's result on fractional matchings of uniform hypergraphs and we show that it is sharp if certain block designs exist. We obtain two sharp results for the case r = s = 2: we prove BRR(K_{1,λ}, K_{2,2}) = 3λ - 2 and that the smallest n for which any edge coloring of K_{λ,n} contains either a monochromatic K_{1,λ} or a multicolored K_{2,2} is λ^{2}.

Original language | English |
---|---|

Pages (from-to) | 101-112 |

Number of pages | 12 |

Journal | Journal of Combinatorial Theory, Series A |

Volume | 113 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2006 |

### Fingerprint

### Keywords

- 3-design
- Extremal configurations
- Mono- and multicolored bipartite graphs
- Rainbow matrix
- Ramsey theory

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science

### Cite this

*Journal of Combinatorial Theory, Series A*,

*113*(1), 101-112. https://doi.org/10.1016/j.jcta.2005.07.003

**Mono-multi bipartite Ramsey numbers, designs, and matrices.** / Balister, Paul N.; Gyárfás, A.; Lehel, Jeno; Schelp, Richard H.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory, Series A*, vol. 113, no. 1, pp. 101-112. https://doi.org/10.1016/j.jcta.2005.07.003

}

TY - JOUR

T1 - Mono-multi bipartite Ramsey numbers, designs, and matrices

AU - Balister, Paul N.

AU - Gyárfás, A.

AU - Lehel, Jeno

AU - Schelp, Richard H.

PY - 2006/1

Y1 - 2006/1

N2 - Eroh and Oellermann defined BRR (G1, G2) as the smallest N such that any edge coloring of the complete bipartite graph KN,N contains either a monochromatic G1 or a multicolored G2. We restate the problem of determining BRR (K1,λ, Kr,s) in matrix form and prove estimates and exact values for several choices of the parameters. Our general bound uses Füredi's result on fractional matchings of uniform hypergraphs and we show that it is sharp if certain block designs exist. We obtain two sharp results for the case r = s = 2: we prove BRR(K1,λ, K2,2) = 3λ - 2 and that the smallest n for which any edge coloring of Kλ,n contains either a monochromatic K1,λ or a multicolored K2,2 is λ2.

AB - Eroh and Oellermann defined BRR (G1, G2) as the smallest N such that any edge coloring of the complete bipartite graph KN,N contains either a monochromatic G1 or a multicolored G2. We restate the problem of determining BRR (K1,λ, Kr,s) in matrix form and prove estimates and exact values for several choices of the parameters. Our general bound uses Füredi's result on fractional matchings of uniform hypergraphs and we show that it is sharp if certain block designs exist. We obtain two sharp results for the case r = s = 2: we prove BRR(K1,λ, K2,2) = 3λ - 2 and that the smallest n for which any edge coloring of Kλ,n contains either a monochromatic K1,λ or a multicolored K2,2 is λ2.

KW - 3-design

KW - Extremal configurations

KW - Mono- and multicolored bipartite graphs

KW - Rainbow matrix

KW - Ramsey theory

UR - http://www.scopus.com/inward/record.url?scp=30444460880&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30444460880&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2005.07.003

DO - 10.1016/j.jcta.2005.07.003

M3 - Article

AN - SCOPUS:30444460880

VL - 113

SP - 101

EP - 112

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -