In this paper we improve on the state-of-the-art time hierarchy for time-bounded Kolmogorov complexity. More precisely, we prove that there are infinitely many $n∈ \mathbf{N}$ such that there are strings of size $n$ which are the output of programs of size $f=o(\log n)$ running in $o(2^fT)$ steps but are not the output of any program of size $f$ running in $T$ steps. The previous gap was exponential in $2^f$. This result is established by studying a new problem in combinatorics: a list $E$ of $pk$ numbers in $[1;N]$ is shared evenly between $p$ players, who each get to output an answer depending on their share of $E$. The players must establish a communication-free strategy ensuring that one of them outputs a number not belonging to $E$.