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On Finite Polarized Partition Relations

Published online by Cambridge University Press:  20 November 2018

V. Chvátal
V. Chvátal*
Affiliation:
University of New Brunswick, Fredericton
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Call an m × n array an m × n; k array if its mn entries come from a set of k elements. An m × n; 1 array has mn like entries. We write

(1)

if every m × n; k array contains a p × q; 1 sub-array. The negation of (1) is written

and means that there is an m × n; k array containing no p × q; 1 sub-array. Relations (1) are called "polarized partition relations among cardinal numbers" by P. Erdös and R. Rado [2]. In this note we prove the following theorems.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 3 , June 1969 , pp. 321 - 326
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Erdös, P., Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53 (1947) 292–299.Google Scholar
2. Erdös, P. and Rado, R., A partition calculus in set theory. Bull. Amer. Math. Soc. 62 (1956) 427–489.Google Scholar
3. Frasnay, C., Partages d'ensembles de parties et de produits d'ensembles. C.R.Acad. Sci. Paris 258 (1964) 1373–1376.Google Scholar
4. Guy, R.K., A problem of Zarankiewicz, Theory of graphs (edited by P. Erdos and G. Katona, Akademiai Kiado, Budapest 1968) 119–150.Google Scholar
5. Zarankiewicz, K., Problem PI 01. Colloq. Math. 2 (1951) 301.Google Scholar