# Products of Luzin-type sets with combinatorial properties

## TL;DR

This paper constructs Luzin-type subsets of the real line with strong combinatorial properties, demonstrating their behavior in finite powers and their non-Menger product, using a new combinatorial approach that works in models where previous methods fail.

## Contribution

It introduces a purely combinatorial method to construct Luzin-type sets with Rothberger property whose product is not Menger, even in models like the Random model.

## Key findings

- Constructed Luzin-type sets with Rothberger property in all finite powers
- Demonstrated non-Menger product of these sets
- Applicable in models where previous methods do not work

## Abstract

We construct Luzin-type subsets of the real line in all finite powers Rothberger, with a non-Menger product. To this end, we use a purely combinatorial approach which allows to weaken assumptions used earlier to construct sets with analogous properties. Our assumptions hold, e.g., in the Random model, where already known category theoretic methods fail.