## 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 L1 \cup L2 \cup ··· \cup Lp of the lines of Rn, there is a partition Rn = S1 \cup S2 \cup ··· \cup Sp such that each line in Li meets Si 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 Li meets Si in a finite set.)
This yields as a corollary a classical result of Sierpinski that 2\omega = \omega1 is equivalent to the statement that there exists a partition R3 = S1 \cup S2 \cup S3 such that for each i for every line l parallel to the i-axis, l \cap Si 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 Rn (n \geq 2) are partitioned into \omega disjoint pieces L = \bigcupi < \omega Li, then there is a partition Rn = \bigcupi < \omega Si such that for each i every line in Li meets Si in a finite set.
A related question about partitions on \omega1 is considered, and using the Axiom of Determinateness (AD), the following result is obtained: (ZF+AD+DC) For every partition [\omega1]2 = \bigcupn < \omega Qn there is a partition \omega1 = \bigcupn < \omega An such that [Ak]2 meets only finitely many Qn for each k < \omega.
This contrasts sharply with a result of Todorcevic that in ZFC there exists a partition [\omega1]2 = \bigcupn < \omega Qn such that for every uncountable A \subset \omega1, [A]2 meets every Qn.