Erdős problem 835

Does there exist a $k>2$ such that the $k$-sized subsets of {1,...,2k} can be coloured with $k+1$ colours such that for every $A\subset \{1,\ldots,2k\}$ with $\lvert A\rvert=k+1$ all $k+1$ colours appear among the $k$-sized subsets of $A$?