Reals in the Matet and Willow Models

TL;DR

This paper explores the regularity properties of Matet and Willow tree forcing notions at the ^1_2 level, filling gaps in the understanding of their implications in descriptive set theory.

Contribution

It establishes the regularity implications for Matet and Willow tree forcings, extending the theory to related locally countable closed graphs.

Findings

Regularity implications for Matet and Willow forcings are established.

Results extend to locally countable closed graphs.

Provides new links in the projective hierarchy at ^1_2 level.

Abstract

In this article, we try to complete the regularity implications between the regularitites of the well-known tree forcing notions at the $\boldsymbol{\Delta}^1_2$ level of the projective hierarchy. The missing links in this case were the regularities corresponding to Matet and Willow tree Forcings. Some of the techniques yield more general results related to locally countable closed graphs too, which we mention as corollaries.