Strong Fréchet properties of spaces constructed from squares and AD families

We answer questions of Arhangel’skiĭ using spaces defined from combinatorial objects. We first establish further convergence properties of a space constructed from ✷(κ) showing it is Fréchet-Urysohn for finite sets and a w-space that is not a W-space. We also show that under additional assumptions it may be not bi-sequential, and so providing a consistent example of an absolutely Fréchet α₁ space that is not bise-quential. In addition, if we do not require the space being α₁, we can construct a ZFC example of a countable absolutely Fréchet space that is not bisequential from an almost disjoint family of subsets of the natural numbers.