Hostname: page-component-848d4c4894-m9kch Total loading time: 0 Render date: 2024-05-22T22:29:51.026Z Has data issue: false hasContentIssue false
  • English
  • Français

Ramsey numbers for certain k-graphs

Part of: Graph theory Designs and configurations

Published online by Cambridge University Press:  09 April 2009

Stefan A. Burr  and
Richard A. Duke
Stefan A. Burr
Affiliation:
Computer Science Department, City College, (CUNY) New York, New York 10031, U.S.A.
Richard A. Duke
Affiliation:
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30322, U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

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.

We are interested here in the Ramsey number r(T, C), where C is a complete k-uniform hypergraph and T is a “tree-like” k-graph. Upper and lower bounds are found for these numbers which lead, in some cases, to the exact value for r(T, C) and to a generalization of a theorem of Chváta1 on Ramsey numbers for graphs. In other cases we show that a determination of the exact values of r(T, C) would be equivalent to obtaining a complete solution to existence question for a certain class of Steiner systems.

MSC classification

Secondary: 05C35: Extremal problems 05B35: Matroids, geometric lattices
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 30 , Issue 3 , February 1981 , pp. 284 - 296
Copyright
Copyright © Australian Mathematical Society 1981

References

Beineke, L. W. and Pippert, R. E. (1971), ‘Properties and characterizations of k trees’, Mathematika 18, 141–151.CrossRefGoogle Scholar
Burr, S. A. (1979), ‘Ramsey numbers involving graphs with long suspended’, to appear.Google Scholar
Burr, S. A. and Erdös, P. (1970), ‘Generalizations of a Ramsey-theoretic result of Chvatál’, to appear.Google Scholar
Burr, S. A., Erdös, P. and Spencer, J. (1975), ‘Ramsey theorems for multiple copies of graphs’, Trans. Amer. Math. Soc. 209, 87–89.CrossRefGoogle Scholar
Chartrand, G., Gould, R. J. and Polimeni, A. D. (1979), ‘On Ramsey numbers of forests versus nearly complete graphs’, Notices Amer. Math. Soc. 26, A–568.Google Scholar
Chvátal, V. (1977), ‘Tree-complete graph Ramsey numbers’, J. Graph Theory, 1, 93.CrossRefGoogle Scholar
DeVries, H. L. (1977), ‘On property B and on Steiner systems’, Math. Z. 153, 155–159.CrossRefGoogle Scholar
Duke, R. A. (1975), ‘Ramsey numbers of families of 2-complexes’, Proc. 6th S-E Conference on Combinatorics, Graph Theory and Computing Florida Atlantic University, Boca Raton, Florida, 1975, pp. 265–277 (Congressus Numerantium, No. XIV, Utilitas Math., Winnipeg, Man.)Google Scholar
Duke, R. A. and Erdös, P. (1977), ‘Systems of finite sets having a common intersection’, Proc. 8th S-E Conference on Combinatorics, Graph Theory and Computing Florida Atlantic University, Boca Raton, Florida, 1977, pp. 247–252. (Congressus Numerantium, No. XIX, Utilities Math., Winnipeg, Man.)Google Scholar
Duke, R. A. and Harary, F. (1976), ‘Generalized Ramsey theory VI: Ramsey numbers for small plexes’, J. Austral. Math. Soc. Ser. A 22, 400–410.CrossRefGoogle Scholar
Faudree, R. J., Schelp, R. H. and Sheehan, J. (1979), ‘Ramsey numbers for matchings’, to appear.CrossRefGoogle Scholar
Frankl, P. (1977), ‘On families of finite sets no two of which intersect in a singleton’, Bull. Austral. Math. Soc. 17, 125–135.CrossRefGoogle Scholar
Kramer, E. S. and Mesner, D. M. (1975), ‘Admissible parameters for Steiner systems S(t, k, v) with a table for all (v – t)< 498’, Utilitas Math. 7, 211–222.Google Scholar
Mendelsohn, N. S. and Hung, S. H. Y. (1972), ‘On the Steiner systems S(3, 4, 14) and S(4, 5, 15)’, Utilitas Math. 1, 5–96.Google Scholar
Stahl, S. (1975), ‘On the Ramsey number of r(F, Km), where F is a forest’, Canad. J. Math. 27, 585–589.CrossRefGoogle Scholar