# Inclusion modulo nonstationary

## TL;DR

This paper extends classical universality results to higher cardinals, showing that under GCH, for any analytic quasi-order, there exists a Lipschitz reduction to a structure defined on $oldsymbol{ ext{kappa}^ ext{kappa}}$ with a stationary subset, in a suitable forcing extension.

## Contribution

It proves a consistency result that generalizes Hechler's theorem to higher uncountable cardinals and stationary subsets, establishing universality of $oldsymbol{ ext{kappa}^ ext{kappa}}$ structures.

## Key findings

- Universality of higher analogues $oldsymbol{ ext{kappa}^ ext{kappa}}$ established under GCH.
- Existence of Lipschitz reductions for all analytic quasi-orders over $oldsymbol{ ext{kappa}^ ext{kappa}}$.
- Consistency result holds in cofinality-preserving GCH-preserving forcing extensions.

## Abstract

A classical theorem of Hechler asserts that the structure $\left(\omega^\omega,\le^*\right)$ is universal in the sense that for any $\sigma$-directed poset P with no maximal element, there is a ccc forcing extension in which $\left(\omega^\omega,\le^*\right)$ contains a cofinal order-isomorphic copy of P. In this paper, we prove a consistency result concerning the universality of the higher analogue $\left(\kappa^\kappa,\le^S\right)$: Theorem. Assume GCH. For every regular uncountable cardinal $\kappa$, there is a cofinality-preserving GCH-preserving forcing extension in which for every analytic quasi-order Q over $\kappa^\kappa$ and every stationary subset S of $\kappa$, there is a Lipschitz map reducing Q to $(\kappa^\kappa,\le^S)$.