Problem 835

Erdős Problem #835 [status: verifiable; formalized: yes]. Does there exist a $k>2$ such that the $k$-sized subsets of $\{1,\ldots,2k\}$ 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 $A$? Current best: The chromatic numbers listed at this website show that this is false for $3\leq k\leq 8$. Prize: no. Tags: graph theory, hypergraphs.