Forcing and the Halpern-L\"auchli Theorem

We investigate the effects of various forcings on several forms of the Halpern-L\"auchli 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-L\"auchli Theorem at $\kappa$. We also show that the Halpern-L\"auchli Theorem is preserved by ${{<}\kappa}$-closed forcings assuming $\kappa$ is measurable, following some observed reflection properties.