  In this short paper we study the game of cops and robbers, which is played on
the vertices of some fixed graph $G$. Cops and a robber are allowed to move
along the edges of $G$ and the goal of cops is to capture the robber. The cop
number $c(G)$ of $G$ is the minimum number of cops required to win the game.
Meyniel conjectured a long time ago that $O(\sqrt{n})$ cops are enough for any
connected $G$ on $n$ vertices. Improving several previous results, we prove
that the cop number of $n$-vertex graph is at most $n 2^{-(1+o(1))\sqrt{\log
n}}$.

Combinatorics