# On the nontrivial solvability of systems of homogeneous linear equations over $\mathbb Z$ in ZFC

## TL;DR

This paper explores the conditions under which uncountable systems of homogeneous linear equations over integers are nontrivially solvable in ZFC, focusing on the compactness property relative to cardinalities.

## Contribution

It investigates the nontrivial solvability of large systems of linear equations over integers within ZFC, extending previous work by Herrlich and Tachtsis.

## Key findings

- Identifies specific uncountable cardinals where compactness holds
- Establishes conditions for nontrivial solutions in large systems
- Provides new insights into the structure of solutions over $\

## Abstract

Following a recent paper by Herrlich and Tachtsis, we investigate in ZFC the following compactness question: for which unountable cardinals $\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb Z$-linear equations is nontrivially solvable in $\mathbb Z$ provided that each its nonempty subsystem of cardinality $<\kappa$ is nontrivially solvable in $\mathbb Z$?