Generalized Ramsey Theory for Multiple Colors
نویسندگان
چکیده
In this paper, we study the generalized Ramsey number r(G, , . . ., Gk) where the graphs GI , . . ., Gk consist of complete graphs, complete bipartite graphs, paths, and cycles. Our main theorem gives the Ramsey number for the case where G 2 , . . ., G,, are fixed and G, ~_C, or P,, with n sufficiently large . If among G2 , . . ., 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 G2 , . . ., Gk 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,, C z , Ck ) and r(C„, CI , C,.,, C) when n is large.
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