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Local and Mean Ramsey Numbers for Trees

✍ Scribed by B. Bollobás; A. Kostochka; R.H. Schelp

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
75 KB
Volume
79
Category
Article
ISSN
0095-8956

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✦ Synopsis

In this note we find the local and mean k-Ramsey numbers for many trees for which the Erdo s So s tree conjecture holds.

2000 Academic Press

The usual Ramsey number R(G, k) is the smallest positive integer n such that any coloring of the edges of K n by at most k colors contains a monochromatic copy of G. Over the past years several papers have been written in which the number of different colors used has no longer been restricted to k, but a restriction is placed on the number of colored edges incident to the vertices. To be precise, let H be a fixed graph of order n and let f be a coloring on the edges of H. For each v # V(H) define k f (v) as the number of distinct colors that appear on the edges of H incident to v. The coloring f is called a local k-coloring if k f (v) k for all v # V(H), and is called a mean k-coloring if (1Ân) v k f (v) k. Further, the local k-Ramsey number R(G, k loc) is defined as the smallest positive integer n such that

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