  We show that the cop number of any graph on 18 or fewer vertices is at most
3. This answers a question posed by Andreae in 1986, as well as more recently
by Baird et al. We also find all 3-cop-win graphs on 11 vertices, narrow down
the possible 4-cop-win graphs on 19 vertices and make some progress on finding
the minimum order of 3-cop-win planar graphs.
