Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\{1,\dots ,n\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$.Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\ge n^{\omega (n)}$ for some $\omega (n)\to \infty$ as $n\to \infty$, and $F(n)<(1+o(1){)}^n$?