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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 809.04004

**Autor: ** Erdös, Paul; Jackson, Steve; Mauldin, R.Daniel

**Title: ** On partitions of lines and space. (In English)

**Source: ** Fundam. Math. 145, No.2, 101-119 (1994).

**Review: ** The main theorem of this paper is:

Suppose \theta is 0 or a limit ordinal, s in \omega, and \theta+s \geq 1. Then 2^{\omega} \leq \omega_{\theta+s} is equivalent to the statement: For every n \geq 2 and partition L_{1} \cup L_{2} \cup ··· \cup L_{p} of the lines of **R**^{n}, there is a partition **R**^{n} = S_{1} \cup S_{2} \cup ··· \cup S_{p} such that each line in L_{i} meets S_{i} in a set of size \leq \omega_{\theta+s - p+1}. (In the case \theta = 0 and s - p+1 < 0, then each line in L_{i} meets S_{i} in a finite set.)

This yields as a corollary a classical result of Sierpinski that 2^{\omega} = \omega_{1} is equivalent to the statement that there exists a partition **R**^{3} = S_{1} \cup S_{2} \cup S_{3} such that for each i for every line l parallel to the i-axis, l \cap S_{i} is finite. It and variations prove generalizations of Sierpinski which are due to Kuratowski, Sikorski, Erd\H os, Davies, Bagemihl, Simms, Bergman, Hrushovski, Galvin, and Gruenhage.

In the case of partitions into infinitely many pieces, their main result is: If the lines L in **R**^{n} (n \geq 2) are partitioned into \omega disjoint pieces L = \bigcup_{i < \omega} L_{i}, then there is a partition **R**^{n} = \bigcup_{i < \omega} S_{i} such that for each i every line in L_{i} meets S_{i} in a finite set.

A related question about partitions on \omega_{1} is considered, and using the Axiom of Determinateness (AD), the following result is obtained: (ZF+AD+DC) For every partition [\omega_{1}]^{2} = \bigcup_{n < \omega} Q_{n} there is a partition \omega_{1} = \bigcup_{n < \omega} A_{n} such that [A_{k}]^{2} meets only finitely many Q_{n} for each k < \omega.

This contrasts sharply with a result of Todorcevic that in ZFC there exists a partition [\omega_{1}]^{2} = \bigcup_{n < \omega} Q_{n} such that for every uncountable A \subset \omega_{1}, [A]^{2} meets every Q_{n}.

**Reviewer: ** A.W.Miller (Madison)

**Classif.: ** * 04A20 Combinatorial set theory

03E60 Axiom of determinacy, etc.

04A30 Continuum hypothesis and generalizations

**Keywords: ** value of continuum; partitions of lines in Euclidean space; axiom of determinateness

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag