Full Souslin trees at small cardinals

Abstract

A $\hat{I}º$-tree is said to be full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $\hat{I}º$-Souslin tree may consistently exist. Shelah gave an affirmative answer of height a strong limit Mahlo cardinal. Here, it is shown that these trees may consistently exist at small cardinals. Indeed, there can be ${\aleph }_3$ many full ${\aleph }_2$-trees such that the product of any countably many of them is an ${\aleph }_2$-Souslin tree.

Relaxed the hypothesis of Theorem C. Added Proposition 2.3