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Partitioning large vector spaces

  • James H. Schmerl (a1)

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The theme of this paper is the generalization of theorems about partitions of the sets of points and lines of finite-dimensional Euclidean spaces ℝ d to vector spaces over ℝ of arbitrary dimension and, more generally still, to arbitrary vector spaces over other fields so long as these fields are not too big. These theorems have their origins in the following striking theorem of Sierpiński [12] which appeared a half century ago.

Sierpiński's Theorem. The Continuum Hypothesis is equivalent to: There is a partition {X, Y, Z} of3 such that if ℓ is a line parallel to the x-axis [respectively: y-axis, z-axis] then X [respectively: Yℓ, Z] is finite.

The history of this theorem and some of its subsequent developments are discussed in the very interesting article by Simms [13]. Sierpiński's Theorem was generalized by Kuratowski [9] to partitions of ℝ n+2 into n + 2 sets obtaining an equivalence with . The geometric character that Sierpiński's Theorem and its generalization by Kuratowski appear to have is bogus, since the lines parallel to coordinate axes are essentially combinatorial, rather than geometric, objects. The following version of Kuratowski's theorem emphasizes its combinatorial character.

Kuratowski's Theorem. Let n < ω and A be any set. ThenA∣ ≤ ℵ n if and only if there is a partition P: A n+2n + 2 such that if in + 1 and ℓ is a line parallel to the i-th coordinate axis, then {x: P(x) = i} is finite.

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[1] Davies, R. O., On a denumerable partition problem of Erdős, Proceedings of the Cambridge Philosophical Society, vol. 59 (1963), pp. 501502.
[2] Davies, R. O., On a problem of Erdős concerning decompositions of the plane, Proceedings of the Cambridge Philosophical Society, vol. 59 (1963), pp. 3336.
[3] Davies, R. O., The power of the continuum and some propositions of plane geometry, Fundamenta Mathematicae, vol. 52 (1963), pp. 277281.
[4] Erdős, P., Some remarks on set theory, III, Michigan Mathematics Journal, vol. 2 (1954), pp. 5157.
[5] Galvin, F., an apparently unpublished result from [7].
[6] Gruenhage, G., Covering properties on X2 ∖ Δ, W-sets, and compact subsets of Σ-products, Topology and its Applications, vol. 17 (1984), pp. 287304.
[7] Komjáth, P., Set theoretic constructions in Euclidean spaces, Chapter XII in New trends in discrete and computational geometry (Pach, J., editor), Springer-Verlag, Berlin-Heidelberg-New York, 1993, pp. 303325.
[8] Komjáth, P., Three clouds cover the plane, Annals of Pure and Applied Logic, vol. 109 (2001), pp. 7175.
[9] Kuratowski, C., Sur une caracterisation des alephs, Fundamenta Mathematicae, vol. 38 (1951), pp. 1417.
[10] Schmerl, J. H., Countable partitions of the sets of points and lines, Fundamenta Mathematicae, vol. 160 (1999), pp. 183196.
[11] Schmerl, J. H., How many clouds cover the plane?, Fundamenta Mathematicae, to appear.
[12] Sierpiński, W., Sur une propriété paradoxale de l'espace à trois dimensions équivalente à l'hypothèse du continu, Rendiconti del Circolo Matematico di Palermo (Serie II), vol. 1 (1952), pp. 710.
[13] Simms, J. C., Sierpiński's theorem, Bulletin of the Belgian Mathematical Society, Simon Stevin, vol. 65 (1991), pp. 69163.

Partitioning large vector spaces

  • James H. Schmerl (a1)

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