## Jan Šaroch

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

Comment.Math.Univ.Carolin. 61,2 (2020) 155-164.**Abstract:**Motivated by the paper by H.\ Herrlich, E.\ Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals~$\kappa$, an arbitrary nonempty system $S$ of homogeneous $\mathbb Z$-linear equations is nontrivially solvable in $\mathbb Z$ provided that each of its subsystems of cardinality less than $\kappa$ is nontrivially solvable in $\mathbb Z$?

**Keywords:** homogeneous $\mathbb Z$-linear equation; $\kappa$-free group; $\mathcal L_{\omega_1\omega}$-compact cardinal

**DOI:** DOI 10.14712/1213-7243.2020.017

**AMS Subject Classification:** 08A45 13C10 20K30 03E35 03E55

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