# Finite powers and products of Menger sets

## TL;DR

This paper constructs specific examples of Menger sets with particular properties regarding their finite powers and products, and extends these results to a broader set-theoretic model using forcing techniques.

## Contribution

It introduces new constructions of Menger sets that are not Scheepers and explores their properties in different set-theoretic contexts.

## Key findings

- Existence of a Menger set not Scheepers under mild hypotheses
- Construction of Menger sets with non-Menger products in all finite powers
- Results hold in the Blass–Shelah model for various ultrafilter and dominating numbers

## Abstract

We construct, using mild combinatorial hypotheses, a real Menger set that is not Scheepers, and two real sets that are Menger in all finite powers, with a non-Menger product. By a forcing-theoretic argument, we show that the same holds in the Blass--Shelah model for arbitrary values of the ultrafilter and dominating number.