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

TL;DR

This paper investigates the Halpern--L"auchli Theorem at uncountable cardinals, proving its validity under certain conditions in ZFC and demonstrating failures of weak forms at other large cardinal levels.

Contribution

It establishes new ZFC results for the theorem at strongly inaccessible cardinals and shows consistency and failure results at various large cardinal and singular cardinal levels.

Findings

Proves the theorem for one tree of height κ in ZFC when κ is strongly inaccessible.

Shows consistency of the theorem for finitely many trees at certain strong limit cardinals.

Demonstrates failures of weak forms at non-Mahlo and singular strong limit cardinals.

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.