# Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

## TL;DR

This paper explores how symmetric extensions based on Lévy Collapse affect dependent choice principles, extending previous results and proving a conjecture related to the preservation of dependent choice in various models.

## Contribution

It extends results on symmetric extensions with Lévy Collapse, proves a conjecture on dependent choice preservation, and analyzes conditions for maintaining dependent choice in different models.

## Key findings

- Dependent choice can be preserved in symmetric extensions under certain conditions.
- Proved a conjecture of Ioanna Dimitriou regarding symmetric extensions.
- Identified conditions involving $vy$-distributivity and strategic closure for preserving dependent choice.

## Abstract

We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if $V$ is a model of ZFC, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-distributive and $\mathcal{F}$ is $\kappa$-complete. Further we observe that if $V$ is a model of ZF + $DC_{\kappa}$, then $DC_{<\kappa}$ can be preserved in the symmetric extension of $V$ in terms of symmetric system $\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle$, if $\mathbb{P}$ is $\kappa$-strategically closed and $\mathcal{F}$ is $\kappa$-complete.