Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-15T03:39:46.748Z Has data issue: false hasContentIssue false
  • English
  • Français

A bipartite Ramsey problem and the Zarankiewicz numbers

Published online by Cambridge University Press:  18 May 2009

Robert W. Irving
Robert W. Irving
Affiliation:
Department of Mathematics, University of Salford, Salford M5 4WT
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.

Beineke and Schwenk [1] have defined the bipartite Ramsey number R(m,n), for integers m, n (1≤m≤n ), to be the smallest integer p such that any 2-colouring of the edges of the complete bipartite graph Kp, v forces the appearance of a monochromatic Km, n. In [1] the following results are established:

with equality if there is a Hadamard matrix of order 2(n−1), n odd,

if there is a Hadamard matrix of order 4(n−1),

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 19 , Issue 1 , January 1978 , pp. 13 - 26
Copyright
Copyright © Glasgow Mathematical Journal Trust 1978

References

REFERENCES

1.Beineke, L. W. and Schwenk, A. J., On a bipartite form of the Ramsey problem, in Proc. Fifth British Combinatorial Conference, Aberdeen, 1975 (Utilitas Mathematica Publishing Inc., Winnipeg), 17–22.Google Scholar
2.Doyen, J., A table of small nondegenerate f-designs without repeated blocks, Typescript.Google Scholar
3.Guy, R. K., A many-facetted problem of Zarankiewicz, in The Many Facets of Graph Theory, Lecture Notes in Mathematics No. 110 (Springer-Verlag, 1969), 129–148.CrossRefGoogle Scholar
4.Hales, A. W. and Jewett, R. I., Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229.CrossRefGoogle Scholar
5.Harary, F., The foremost open problems in generalised Ramsey theory, in Proc. Fifth British Combinatorial Conference, Aberdeen, 1975 (Utilitas Mathematica Publishing Inc., Winnipeg), 269–282.Google Scholar
6.Zarankiewicz, K., Problem P101, Colloq. Math., 2 (1951) 301.Google Scholar