Base matrices of various heights

Abstract

AbstractA classical theorem of Balcar, Pelant, and Simon says that there is a base matrix of height ${\mathfrak h}$ , where ${\mathfrak h}$ is the distributivity number of ${\cal P} (\omega ) / {\mathrm {fin}}$ . We show that if the continuum ${\mathfrak c}$ is regular, then there is a base matrix of height ${\mathfrak c}$ , and that there are base matrices of any regular uncountable height $\leq {\mathfrak c}$ in the Cohen and random models. This answers questions of Fischer, Koelbing, and Wohofsky.