Rules and Reals
Martin Goldstern, Menachem Kojman
Abstract
A ``k-rule" is a sequence A=((An,Bn):n<omega) of pairwise disjoint sets Bn, each of cardinality at most k, where An is a subset of Bn. A set X of natural numbers (a ``real'') follows a rule A if for infinitely many n we have that the intersection of X with Bn is exactly An. There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all k-rules, sk, and the least number of k-rules without a real following all of them, rk. We investigate these cardinal invariants and their connection to some well-known cardinals from Cichon's diagram. The original motivation for discovering rules was an attempt to construct a maximal homogeneous family over omega. The consistency of such a family is still open.
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