Geodesic
On infinite partitions of lines and space
Fundamenta Mathematicae, Tome 152 (1997) no. 1, pp. 75-95 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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Given a partition P:L → ω of the lines in $ℝ^n$, n ≥ 2, into countably many pieces, we ask if it is possible to find a partition of the points, $Q:ℝ^n → ω$, 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.
DOI : 10.4064/fm_1997_152_1_1_75_95
Keywords: transfinite recursion, Martin's Axiom, forcing, geometry, infinite partitions 
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P. Erdős; Steve Jackson; Daniel Mauldin. On infinite partitions of lines and space. Fundamenta Mathematicae, Tome 152 (1997) no. 1, pp. 75-95. doi: 10.4064/fm_1997_152_1_1_75_95

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