Rules and Reals
Abstract
A ``k-rule" is a sequence A=((A_n,B_n):n<omega) of pairwise disjoint sets B_n, each of cardinality at most k, where A_n is a subset of B_n. 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 B_n is exactly A_n. There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all k-rules, s_k, and the least number of k-rules without a real following all of them, r_k. 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.
Cite
@article{arxiv.math/9707204,
title = {Rules and Reals},
author = {Martin Goldstern and Menachem Kojman},
journal= {arXiv preprint arXiv:math/9707204},
year = {2016}
}