Full Souslin trees at small cardinals

Abstract

A ‐tree is full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full ‐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 many full ‐trees such that the product of any countably many of them is an ‐Souslin tree.