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/logn), and r(C 2m , C 2m , K n )≤c(n/logn)m/(m−1) for sufficiently large n.
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Received: July 25, 2000 Final version received: July 30, 2002
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ID="*" Partially supported by RGC, Hong Kong; FRG, Hong Kong Baptist University; and by NSFC, the scientific foundations of education ministry of China, and the foundations of Jiangsu Province
Acknowledgments. The authors are grateful to the referee for his valuable comments.
AMS 2000 MSC: 05C55
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Shiu, W., Lam, P. & Li, Y. On Some Three-Color Ramsey Numbers. Graphs and Combinatorics 19, 249–258 (2003). https://doi.org/10.1007/s00373-002-0495-7
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DOI: https://doi.org/10.1007/s00373-002-0495-7