Subgraphs of random graphs

Authors:
D. H. Fremlin and M. Talagrand

Journal:
Trans. Amer. Math. Soc. **291** (1985), 551-582

MSC:
Primary 60C05; Secondary 05C80

DOI:
https://doi.org/10.1090/S0002-9947-1985-0800252-6

MathSciNet review:
800252

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be an undirected graph on , and let . Following P. Erdös and A. Hajnal, we write to mean: whenever is a measurable set of Lebesgue measure at least for every , then there is some such that appears in the graph in the sense that there is a strictly increasing function such that whenever . We give an algorithm for determining when for finite , and we show that for infinite if there is a such that for every finite . Our results depend on a new condition, expressed in terms of measures on , sufficient to imply that appears in (Theorem 2F), and enable us to identify the extreme points of some convex sets of measures (Theorem 5H).

**[1]**P. Erdős and A. Hajnal,*Some remarks on set theory. IX. Combinatorial problems in measure theory and set theory*, Michigan Math. J.**11**(1964), 107–127. MR**0171713**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1985-0800252-6

Keywords:
Random graphs,
,
extreme points

Article copyright:
© Copyright 1985
American Mathematical Society