Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 095.03902
Autor: Erdös, Pál; Hajnal, András
Title: Some remarks on set theory. VIII. (In English)
Source: Mich. Math. J. 7, 187-191 (1960).
Review: The authors consider independent sets and graphs (cf. also Erdös-Fodor, Zbl 078.04203). Let R denote the set of real numbers; for every x in R let S(x) be such that x \not in S(x) \subset R. A subset S \subset R is independent provided for every x,y in S, x \ne y one has x \not in S(y), y \not in S(x). Let H0 denote the statement: R can be well-ordered into a \Omegac-sequence such that every set which is not cofinal with \Omegac has measure 0.
Theorem 1: If S(x) (x in R) is of measure 0 and is not everywhere dense, there exist 2 real independent numbers x \ne y (under H0 there are no 3 independent real numbers).
Theorem 2: If S(x) is bounded and has the exterior measure \leq 1, then there are n independent real numbers, for every 1 < n < \omega0. A \sigma-ideal I of subsets R is said to have the property P, symbolically I in P, provided it contains a transfinite sequence B\beta (\beta < \Omegac) of members such that every member of I is contained in some B\beta.
Theorem 3: If \aleph1 = c and I in P, then each graph GR on R contains an infinite chain or an antichain that is not in I (the statement may not hold provided I\not in P).
Theorem 5: Let m < c. Let I\alpha (\alpha < \Omegac) be a sequence of \sigma-ideals of subsets of R, each with property P. Then every graph GR contains, for every n < \omega, a subgraph {xi} \cup {y\nu } (1 < i \leq n, 1 < \alpha < \Omegac) such that (xi, y\alpha) is connected or there is an antichain in GR which is contained in no I\alpha. The authors ask whether theorem 5 holds for m = c; they conjecture also that theorem 5 may not hold if the property P is delated, even for n = m = 2.
Reviewer: G.Kurepa
Classif.: * 04A99 Miscellaneous topics in set theory
Index Words: set theory
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