Countable partitions of the sets of points and lines

The following theorem is proved, answering a question raised by Davies in 1963. If L0 ∪L1 ∪L2 ∪ . . . is a partition of the set of lines of R, then there is a partition R = S0 ∪ S1 ∪ S2 ∪ . . . such that |` ∩ Si| ≤ 2 whenever ` ∈ Li. There are generalizations to some other, higher-dimensional subspaces, improving recent results of Erdős, Jackson & Mauldin. 0. Introduction. A series of papers, beginning in 1919 with Sierpiński [7] and ending with the 1963 paper of Davies [1], culminates in the following theorem. Simms [8] presents a detailed and fascinating account of the history of this and related theorems. Theorem 0.1 (Davies). If m < ω, then the following are equivalent : (1) 2א0 ≤ אm; (2) if 2 ≤ n < ω and L = L0 ∪ L1 ∪ . . . ∪ Lm+1 is a partition of the set of lines of R, then there is a partition R = S0 ∪S1 ∪ . . .∪Sm+1 such that for each ` ∈ Li, ` ∩ Si is finite; (3) there are pairwise nonparallel lines d0, d1, . . . , dm+1 in R2 and a partition R2 = S0 ∪ S1 ∪ . . . ∪ Sm such that if a line ` is parallel to di, then ` ∩ Si is finite. Prior to the publication of this result, yet motivated by some similar results on finite partitions, Erdős [3] had asked about infinite partitions: If L = L0∪L1∪L2∪. . . is a countable partition of the set of lines of R2, is there a partition R2 = S0∪S1∪S2∪. . . such that for each ` ∈ Li, |`∩Si| ≤ 1? This question was answered negatively by Davies [2]. However, on the positive side, Davies [2] proved the following two closely related theorems. 1991 Mathematics Subject Classification: 03E05, 04A20.