Atlas: Sum and Difference Free Partitions of Vector Spaces by Krzysztof Ciesielski

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The Eleventh Summer Conference on General Topology and Applications
August 10-13, 1995
University of Southern Maine
Gorham, ME, USA

Organizers
J. Baumgartner, D. Briggs, J. deBakker, B. Flagg, G. Gruenhage, M. Guay, Y. Kong, R. Kopperman, S. Shore, J. Rutten, J. Vaughan

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Sum and Difference Free Partitions of Vector Spaces
by
Krzysztof Ciesielski
West Virginia University

Let V stand for a vector space over Q.

For a cardinal \kappa and a partition P of S subset V we say that P is \kappa-sum-free if for every a in V the equation x + y = a has less than \kappa many solutions with x and y from the same element P of the partition P. (We consider the solutions <x, y > and <y, x > identical.)

We say that S subset V is \kappa-sum-free if P = S is \kappa-sum-free. In particular, if partition P of S is \kappa-sum-free then P partitions S into \kappa-sum-free sets.

Similarly we say that partition P of S is \kappa-difference-free if for every a in V, a =/= 0, the equation x - y = a has less than \kappa many solutions with x and y from the same element of partition.

Set S is \kappa-difference-free if P = S is \kappa-difference-free.

We will examine the following cardinal invariants, in which P is countable.

[^(\sigma)](\kappa) = min { |V| : there is no \kappa-sum-free partition P of V }

\sigma(\kappa) = min { |V| : there is no partition P of V into \kappa-sum-free sets }

[^(\delta)](\kappa) = min { |V| : there is no \kappa-difference-free partition P of V }

\delta(\kappa) = min { |V| : there is no partition P of V into \kappa-difference-free sets }

We will concentrate on the cases when \kappa <= \omega.

Date received: April 12, 1996