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Graph Theory and Probability

Published online by Cambridge University Press:  20 November 2018

P. Erdös
P. Erdös*
Affiliation:
University of Toronto and Technion, Haifa
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A well-known theorem of Ramsay (8; 9) states that to every n there exists a smallest integer g(n) so that every graph of g(n) vertices contains either a set of n independent points or a complete graph of order n, but there exists a graph of g(n) — 1 vertices which does not contain a complete subgraph of n vertices and also does not contain a set of n independent points. (A graph is called complete if every two of its vertices are connected by an edge; a set of points is called independent if no two of its points are connected by an edge.) The determination of g(n) seems a very difficult problem; the best inequalities for g(n) are (3)

It is not even known that g(n)1/n tends to a limit. The lower bound in (1) has been obtained by combinatorial and probabilistic arguments without an explicit construction.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 34 - 38
Copyright
Copyright © Canadian Mathematical Society 1959

References

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