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An upper bound on the Ramsey numbers R(3, k)

✍ Scribed by Jerrold R Griggs

Publisher
Elsevier Science
Year
1983
Tongue
English
Weight
372 KB
Volume
35
Category
Article
ISSN
0097-3165

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## Abstract Harary stated the conjecture that for any graph __G__ with __n__ edges and without isolated vertices __r__(__K__~3~,__G__) β©½ 2__n__ + 1. ErdΓΆs, Faudree, Rousseau, and Schelp proved that __r__(__K__~3~,__G__) β©½ ⌈8/3__n__βŒ‰. Here we prove that __r__(__K__~3~,__G__) β©½ ⌊5/2__n__βŒ‹ βˆ’1 for __n_

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The Ramsey number r(H, G) is defined as the minimum N such that for any coloring of the edges of the N-vertex complete graph KN in red and blue, it must contain either a ted H or a blue G. In this paper we show that for any graph G without isolated vertices, r(K,, G)< 2qf 1 where G has q edges. In o

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In this paper we show that for n β‰₯ 4, R(3, 3, . . . , 3) < n!( e-e -1 + 3 2 ) + 1. Consequently, a new bound for Schur numbers is also given. Also, for even n β‰₯ 6, the Schur number S n is bounded by S n < n!( e-e -1 + 3 2 ) -n + 2.