Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T18:09:46.882Z Has data issue: false hasContentIssue false
  • English
  • Français

Graph Theory and Probability

Published online by Cambridge University Press:  20 November 2018

P. Erdös
P. Erdös*
Affiliation:
University of Toronto and Technion, Haifa
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

1. Descartes, Blanche, A three colour problem, Eureka (April, 1947). (Solution March, 1948.)Google Scholar
2. Descartes, Blanche. Solution to Advanced Problem no. 4526, Amer. Math. Monthly, 61 (1954), 352.Google Scholar
3. Erdös, P., Some remarks on the theory of graphs, B.A.M.S. 53, (1947), 292-4.Google Scholar
4. Remarks on a theorem of Ramsey, Bull. Research Council of Israel, Section F, 7 (1957).Google Scholar
5. Erdös, P. and Szekeres, G., A combinatorial problem in geometry, Compositio Math. 2 (1935) 463-70.Google Scholar
6. Kelly, J. B. and Kelly, L. M., Paths and circuits in critical graphs, Amer. J. Math., 76 (1954), 786-92.Google Scholar
7. Mycielski, J., Sur le colorage des graphs, Colloquium Math. 3 (1955), 161-2.Google Scholar
8. Ramsay, F. P., Collected papers, 82-111.Google Scholar
9. Skolem, T., Ein kombinatonscher Satz mit Anwendung auf ein logisches Entscheidungs problem Fund. Math. 20 (1933), 254-61.Google Scholar