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Miklós Schweitzer 1960- Problem 4

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November 18, 2015
college contestsreal analysisset theoryMiklos Schweitzer

Problem Statement

4. Let (Hα)\left (H_{\alpha} \right ) be a system of sets of integers having the property that for any α1≠α2,Hα1∩Hα2\alpha _1 \neq \alpha _2 , H_{\alpha _1}\cap H_{\alpha _2} is a finite set and Hα1≠Hα2H_{{\alpha} _1} \neq H_{{\alpha} _2}. Prove that there exists a system (Hα)\left (H_{\alpha} \right ) of this kind whose cardinality is that of the continuum. Prove further that if none of the intersections of two sets HαH_\alpha contains more than KK elements, then the system (Hα)\left (H_{\alpha} \right ) is countable (KK is an arbitrary fixed integer). (St. 4)
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