#### Document Type

Journal Article

#### Department/Unit

Department of Mathematics

#### Language

English

#### Abstract

In this paper we study three-color Ramsey numbers. Let *K* *i,j* denote a complete *i* by *j*bipartite graph. We shall show that (i) for any connected graphs *G* 1, *G* 2 and *G* 3, if *r*(*G* 1,*G* 2)≥*s*(*G* 3), then *r*(*G* 1, *G* 2, *G* 3)≥(*r*(*G* 1, *G* 2)−1)(χ(*G* 3)−1)+*s*(*G* 3), where *s*(*G* 3) is the chromatic surplus of *G* 3; (ii) (*k*+*m*−2)(*n*−1)+1≤*r*(*K* *1,k* , *K* *1,m* , *K* *n* )≤ (*k*+*m*−1)(*n*−1)+1, and if*k* or *m* is odd, the second inequality becomes an equality; (iii) for any fixed *m*≥*k*≥2, there is a constant *c* such that *r*(*K* *k,m* , *K* *k,m* , *K* *n* )≤*c*(*n*/log*n*), and *r*(*C* *2m* , *C* *2m* , *K* *n* )≤*c*(*n*/log*n*) *m/(m−1)* for sufficiently large *n*.

#### Keywords

Monochromatic graph, Three-color Ramsey number

#### Publication Date

2003

#### Source Publication Title

Graphs and Combinatorics

#### Volume

19

#### Issue

2

#### Start Page

249

#### End Page

258

#### Publisher

Springer Verlag

#### Peer Reviewed

1

#### DOI

10.1007/s00373-002-0495-7

#### Link to Publisher's Edition

http://dx.doi.org/10.1007/s00373-002-0495-7

#### ISSN (print)

14355914

#### Recommended Citation

Shiu, Wai Chee,
Peter Che Bor Lam,
and
Yusheng Li.
"On some three-color Ramsey numbers."
*Graphs and Combinatorics*
19.2
(2003): 249-258.