# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Ramsey theorems for multiple copies of graphsHTML articles powered by AMS MathViewer

by S. A. Burr, P. Erdős and J. H. Spencer
Trans. Amer. Math. Soc. 209 (1975), 87-99 Request permission

## 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 \${K_p}\$ are 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 \leqslant r(mG,nH) \leqslant N + C\$, where \$N = km + ln - min(mi,mj)\$ and where \$C\$ is an effectively computable function of \$G\$ and \$H\$. The method used permits exact evaluation of \$r(mG,nH)\$ for various choices of \$G\$ and \$H\$, especially when \$m = n\$ or \$G = H\$. In particular, \$r(m{K_3},n{K_3}) = 3m + 2n\$ when \$m \geqslant n,m \geqslant 2\$.
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