2 problems

Let $d\ge 2$ be finite. Let $\operatorname{HL}(d,\sigma,\kappa)$ denote the Halpern–Läuchli property for $d$ many regular trees at a measurable cardinal $\kappa$, and let the compa…

Let $de 2$ be finite, let $appa$ be measurable, and let $\sigma<appa$. Write $\operatorname{HL}(d,\sigma,appa)$ for the Halpern–Läuchli property for $d$ many regular trees of heigh…