SQUARES AND NARROW SYSTEMS

A narrow system is a combinatorial object introduced byMagidor and Shelah in connection with work on the tree property at successors of singular cardinals. In analogy to the tree property, a cardinal kappa satisfies the narrow system property if every narrow system of height kappa has a cofinal branch. In this paper, we study connections between the narrow system property, square principles, and forcing axioms. We prove, assuming large cardinals, both that it is consistent that. aleph(omega+1) satisfies the narrow system property and square(aleph omega),(<aleph omega) holds and that it is consistent that every regular cardinal satisfies the narrow system property. We introduce natural strengthenings of classical square principles and show how they can be used to produce narrow systems with no cofinal branch. Finally, we show that the Proper Forcing Axiomimplies that every narrow system of countable width has a cofinal branch but is consistent with the existence of a narrow system of width omega(1) with no cofinal branch.