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**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 H_{0} denote the statement: R can be well-ordered into a \Omega_{c}-sequence such that every set which is not cofinal with \Omega_{c} 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 H_{0} 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 < \omega_{0}. 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 < \Omega_{c}) of members such that every member of I is contained in some B_{\beta}.

Theorem 3: If \aleph_{1} = c and I in P, then each graph G_{R} 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 < \Omega_{c}) be a sequence of \sigma-ideals of subsets of R, each with property P. Then every graph G_{R} contains, for every n < \omega, a subgraph **{**x_{i}**}** \cup **{**y_{\nu} **}** (1 < i \leq n, 1 < \alpha < \Omega_{c}) such that (x_{i}, y_{\alpha}) is connected or there is an antichain in G_{R} 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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