The Halpern--L\"{a}uchli Theorem at singular cardinals and failures of weak versions

Abstract

This paper continues a line of investigation of the Halpern--L\"{a}uchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--L\"{a}uchli Theorem for one tree of height $\kappa$ holds whenever $\kappa$ is strongly inaccessible and the coloring takes less than $\kappa$ colors. We prove consistency of the Halpern--L\"{a}uchli Theorem for finitely many trees of height $\kappa$, where $\kappa$ is a strong limit cardinal of countable cofinality. On the other hand, we prove failure of weak forms of Halpern--\Lauchli\ for trees of height $\kappa$, whenever $\kappa$ is a strongly inaccessible, non-Mahlo cardinal or a singular strong limit cardinal with cofinality the successor of a regular cardinal. We also prove failure in $L$ of a weak version for all strongly inaccessible, non-weakly compact cardinals.