On infinite partitions of lines and space

Given a partition P : L → ω of the lines in R, n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, Q : R → ω, so that each line meets at most m points of its color. Assuming Martin’s Axiom, we show this is the case for m ≥ 3. We reduce the problem for m = 2 to a purely finitary geometry problem. Although we have established a very similar, but somewhat simpler, version of the geometry conjecture, we leave the general problem open. We consider also various generalizations of these results, including to higher dimension spaces and planes. 1. The m-point property for m ≥ 3. We consider here several questions concerning infinite partitions of lines, planes, etc. in R, in particular, colorings of R with prescribed intersection sizes for the lines and points of a given “color”. We are particularly concerned with questions which relate set-theoretic partition properties with the underlying geometry of lines, points, etc., in R. The results presented here extend some of those of [2], answer some of the questions raised there, and introduce some new questions as well. In particular, these results lead to some interesting connections between set-theoretic partition questions and purely geometric questions. Throughout, we use the notions of a partition of a set, A = A0 ∪ A1 ∪ A2 ∪ . . . , and a coloring of the set, f : A → ω, interchangeably. 1991 Mathematics Subject Classification: 03E05, 03E50.