Erdős Problem #1183 — Large monochromatic sublattices of the Boolean lattice

In every two-coloring of the subsets of an $n$-element set, how large a monochromatic family must exist that is closed under both unions and intersections? How large if only closure under unions is required? For the latter quantity $F(n)$, is $F(n)>n^c$ for every fixed $c$ and $F(n)<(1+ε)^n$ for every fixed $ε>0$, once $n$ is sufficiently large?