Generalized Ramsey theory for multiple colors

Abstract

In this paper, we study the generalized Ramsey number r( G 1,…, G k ) where the graphs G 1,…, G k consist of complete graphs, complete bipartite graphs, paths, and cycles. Our main theorem gives the Ramsey number for the case where G 2,…, G k are fixed and G 1 ⋍ C n or P n with n sufficiently large. If among G 2,…, G k there are both complete graphs and odd cycles, the main theorem requires an additional hypothesis concerning the size of the odd cycles relative to their number. If among G 2,…, G k there are odd cycles but no complete graphs, then no additional hypothesis is necessary and complete results can be expressed in terms of a new type of Ramsey number which is introduced in this paper. For k = 3 and k = 4 we determine all necessary values of the new Ramsey number and so obtain, in particular, explicit and complete results for the cycle Ramsey numbers r( C n , C l , C k ) and r( C n , C l , C k , C m ) when n is large.