Generalized Ramsey theory for multiple colors

P. Erdös, R. J. Faudree, C. C. Rousseau, R. H. Schelp

Research output: Contribution to journalArticle

17 Citations (Scopus)

Abstract

In this paper, we study the generalized Ramsey number r(G1,..., Gk) where the graphs G1,..., Gk consist of complete graphs, complete bipartite graphs, paths, and cycles. Our main theorem gives the Ramsey number for the case where G2,..., Gk are fixed and G1 {reversed tilde equals} Cn or Pn with n sufficiently large. If among G2,..., Gk 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(Cn, Cl, Ck) and r(Cn, Cl, Ck, Cm) when n is large.

Original languageEnglish
Pages (from-to)250-264
Number of pages15
JournalJournal of Combinatorial Theory, Series B
Volume20
Issue number3
DOIs
Publication statusPublished - Jun 1976

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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