# Article

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