Erdős Problem #835 — Coloring $k$-Subsets of a $2k$-Set

Does there exist an integer $k>2$ and a coloring of all $k$-element subsets of a $2k$-element set with $k+1$ colors such that, for every $(k+1)$-element subset $A$, the $k$-element subsets of $A$ use all $k+1$ colors?