Forcing and the Halpern-Läuchli Theorem

Abstract

We investigate the effects of various forcings on several forms of the Halpern-Läuchli Theorem. For inaccessible $\kappa$, we show they are preserved by forcings of size less than $\kappa$. Combining this with work of Zhang in \cite{Zhang17} yields that the polarized partition relations associated with finite products of the $\kappa$-rationals are preserved by all forcings of size less than $\kappa$ over models satisfying the Halpern-Läuchli Theorem at $\kappa$. We also show that the Halpern-Läuchli Theorem is preserved by $<\kappa$-closed forcings assuming $\kappa$ is measurable, following some observed reflection properties.

19 pages

Cited by in corpus (1)

• A tail cone version of the Halpern-Läuchli theorem at a large cardinal