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International Journal of Mathematics and Mathematical Sciences
Volume 9, Issue 2, Pages 273-276
http://dx.doi.org/10.1155/S0161171286000339

Generalized Ramsey numbers for paths in 2-chromatic graphs

1Mathematics Department, The University of Toledo, Toledo 43606, Ohio, USA
2Computer Systems Department, The University of Toledo, Toledo 43606, Ohio, USA

Received 26 April 1984; Revised 6 January 1985

Copyright © 1986 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Chung and Liu have defined the d-chromatic Ramsey number as follows. Let 1dc and let t=(cd). Let 1,2,,t be the ordered subsets of d colors chosen from c distinct colors. Let G1,G2,,Gt be graphs. The d-chromatic Ramsey number denoted by rdc(G1,G2,,Gt) is defined as the least number p such that, if the edges of the complete graph Kp are colored in any fashion with c colors, then for some i, the subgraph whose edges are colored in the ith subset of colors contains a Gi. In this paper it is shown that r23(Pi,Pj,Pk)=[(4k+2j+i2)/6] where ijk<r(Pi,Pj), r23 stands for a generalized Ramsey number on a 2-colored graph and Pi is a path of order i.