Ramsey theorems for multiple copies of graphs

S. A. Burr, P. Erdös, Joel Spencer

Research output: Contribution to journalArticle

Abstract

If G and H are graphs, define the Ramsey number r(G, H) to be the least number p such that if the edges of the complete graph Kpare colored red and blue (say), either the red graph contains G as a subgraph or the blue graph contains H. Let mG denote the union of m disjoint copies of G. The following result is proved: Let G and H have k and l points respectively and have point independence numbers of i and j respectively. Then N - 1 ≤ r(mG, nH) ≤ N + C, where N = km + In - min (mi, mj) and where C is an effectively computable function of G and H. The method used permits exact evaluation of r(mGt nH) for various choices of G and H, especially when m — n or G-H. In particular, r(mK3, nK3) = 3m + 2n when m ≥ n, m ≥ 2.

Original languageEnglish (US)
Pages (from-to)87-99
Number of pages13
JournalTransactions of the American Mathematical Society
Volume209
DOIs
StatePublished - 1975

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Ramsey's Theorem
Graph in graph theory
Ramsey number
Independence number
Complete Graph
Subgraph
Disjoint
Union
Denote
Evaluation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Ramsey theorems for multiple copies of graphs. / Burr, S. A.; Erdös, P.; Spencer, Joel.

In: Transactions of the American Mathematical Society, Vol. 209, 1975, p. 87-99.

Research output: Contribution to journalArticle

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