5 problems

Chang's Conjecture.

The cube-in-cube conjecture. If

Let {\mspace{2mu}\mid\mspace{-12mu} {\raise0.6em\text{bullet}}} denote the stick principle, and let Theorem be the theorem discussed in the source, whose hypothesis involves th…

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…

For a partial order type $\psi$ and a cardinal $\kappa$, write $P\nrightarrow(\psi)^1_\kappa$ when some coloring $c:P\to\kappa$ has no monochromatic subset of order type $\psi$; th…