Full Souslin trees at small cardinals

TL;DR

This paper demonstrates the consistent existence of full - ext{Souslin} trees at small cardinals, specifically _2, with properties that their countable products remain _2- ext{Souslin} trees, addressing a question by Kunen.

Contribution

It proves the consistent existence of full - ext{Souslin} trees at _2, with a large family whose products preserve the _2- ext{Souslin} property, at small cardinals.

Findings

Existence of _2- ext{full} _2- ext{Souslin} trees is consistent.

There can be _3 many such trees with product properties.

The product of countably many such trees remains _2- ext{Souslin}.

Abstract

A $\kappa$-tree is said to be full if each of its limit levels omits no more than one potential branch. Kunen asked whether a full $\kappa$-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.