2022/09/22 by Natasha Dobrinen, Saharon Shelah, Dobrinen, Natasha +1
Mathematics · Computer Science · #Advanced Topology and Set Theory #Topological and Geometric Data Analysis #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.2209.11226
This paper continues a line of investigation of the Halpern--Läuchli Theorem at uncountable cardinals. We prove in ZFC that the Halpern--Läuchli Theorem for one tree of height κ holds whenever κ is strongly inaccessible and the coloring takes less than κ colors. We prove consistency of the Halpern--Läuchli Theorem for finitely many trees of height κ, where κ 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 κ, whenever κ 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.