Do Reichenbachian Common Cause Systems of Arbitrary Finite Size Exist?

The principle of common cause asserts that positive correlations between causally unrelated events ought to be explained through the action of some shared causal factors. Reichenbachian common cause systems are probabilistic structures aimed at accounting for cases where correlations of the aforesaid sort cannot be explained through the action of a single common cause. The existence of Reichenbachian common cause systems of arbitrary finite size for each pair of non-causally correlated events was allegedly demonstrated by Hofer-Szab\'o and R\'edei in 2006. This paper shows that their proof is logically deficient, and we propose an improved proof.