Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Subgraphs of random graphs
HTML articles powered by AMS MathViewer

by D. H. Fremlin and M. Talagrand PDF
Trans. Amer. Math. Soc. 291 (1985), 551-582 Request permission

Abstract:

Let $\Delta \subseteq {[\omega ]^2}$ be an undirected graph on $\omega$, and let $u \in [0, 1]$. Following P. Erdös and A. Hajnal, we write $(\omega , 2, u) \Rightarrow \Delta$ to mean: whenever ${E_1} \subseteq [0, 1]$ is a measurable set of Lebesgue measure at least $u$ for every $I \in {[\omega ]^2}$, then there is some $t \in [0, 1]$ such that $\Delta$ appears in the graph ${\Gamma _t} = \{ I: t \in {E_I}\}$ in the sense that there is a strictly increasing function $f: \omega \to \omega$ such that $\{ f(i), f(j)\} \in {\Gamma _t}$ whenever $\{ i, j\} \in \Delta$. We give an algorithm for determining when $(\omega , 2, u) \Rightarrow \Delta$ for finite $\Delta$, and we show that for infinite $\Delta , (\omega , 2, u) \Rightarrow \Delta$ if there is a $\upsilon < u$ such that $(\omega , 2, \upsilon ) \Rightarrow {\Delta ^\prime }$ for every finite $\Delta ’ \subseteq \Delta$. Our results depend on a new condition, expressed in terms of measures on $\beta \omega$, sufficient to imply that $\Delta$ appears in $\Gamma$ (Theorem 2F), and enable us to identify the extreme points of some convex sets of measures (Theorem 5H).
References
  • 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 171713
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 60C05, 05C80
  • Retrieve articles in all journals with MSC: 60C05, 05C80
Additional Information
  • © Copyright 1985 American Mathematical Society
  • 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