Two subsets and their union have same color

Question

Color all nonempty subsets of $[n] = \{1,2,\ldots,n\}$ with colors $1,2,\ldots,r$. Show that, for all large enough $n$, there exist two disjoint nonempty subsets $A,B \subseteq [n]$ such that $A$, $B$, and $A \cup B$ all have the same color.

Answer 1

For two colors, let $n$ be $9$. There have to be five singletons of the same color. Let it be red. Now for this to fail, all pairs of these must be blue. Then all four element sets are red, all three element sets are blue. Then the set of all five must match one pair of its subsets. Can you see how to extend this to $r$ colors?