Sum and difference free partitions of vector spaces

by

Krzysztof Ciesielski

Colloq. Math. 71 (1996), 263-271.

Let V stand for a vector space over rationals Q and S for a subset of V. All the partitions P of a subset S of V will be countable, i.e., the cardinality |P| of P does not exceeds omega. For a cardinal kappa>0 and a partition P of S (subset of V) we say that P is kappa sum free if for every a from V the equation x+y=a has less than kappa many solutions with x and y from the same element of the partition P. We consider the solutions (x,y) and (y,x) identical and ignore the solution (x,y)=(a/2,a/2). We say that a set S is kappa sum free if P={S} is kappa sum free. In particular, if a partition P of S is kappa sum free then P partitions S into kappa sum free sets. Similarly, we say that a partition P of S is kappa difference free if for every non zero a from V the equation x-y=a has less than kappa many solutions (x,y), with x and y from the same element of the partition P. A set S is kappa difference free if P={S} is kappa difference free.

The paper is devoted to study the following cardinal invariants for the case of kappa between 2 and omega.

• 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}.

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