A paopm graph G has no isolated points. I t s R m e y r u m b a r ( G ) i s the m i n i m p such that every 2-coloring of the edges of K contains a monochromatic G. The Ramhey m & t @ m y R(G) i s P the r (G) ' With j u s t one exception, namely Kq, we determine R(G) f o r proper graphs u i t h a t most 4 points. that R = 2n when n i s odd and R = 1 when n i s even. We conclude with the conjecture that f o r a proper graph, R(G) = 1 i f and only i f G = K or Kl with n even.
2 ,n number of monochromatic G i n any 2-coZoring o f K
For the stars Kl n, it i s shajn ,
1. CONCEPTS
The Ramsey nwnber r ( F ) o f a proper graph F (without i s o l a t e d p o i n t s ) i s t h e smallest number p such t h a t i n any 2-coloring of ( t h e l i n e s o f ) t h e complete graph K , s a y green and red without loss of g e n e r a l i t y , t h e r e o c c u r s a monochromatic F. Except f o r new concepts, w e follow t h e n o t a t i o n and terminology of t h e book 171. The Ramsey number r ( F ) of a p r o p e r graph F w a s f i r s t d e f i n e d by Chvdtal and-Harary i n t h e p r e v i o u s papers i n t h i s series [1,2,31 . number o f monochromatic F i n any 2-coloring o f K P W e now d e f i n e t h e Ramsey multiplicity R(F) as t h e minimum r ( F ) A Ramsey realization of F i s a 2-coloring o f K which r (F) produces e x a c t l y R(F) monochromatic F. Two Ramsey r e a l i z a t i o n s Networks 4: 163-173 @ 1974 by John Wiley & Sons, Inc.