Abstract: A topological group G admits a free action if there is a compact Hausdorff space X and an action of G on X such that every non-identity g∈ G acts without fixed points. By Veech's theorem, all locally compact groups admit a free action. On the other side of the spectrum, there are groups that have a common fixed point under any action; such groups are called extremely amenable. We ask which groups other than locally compact admit free actions and how far the property that no non-identity element acts freely under any action is from extreme amenability. These questions may be reformulated in terms of finite colourability of certain graphs. This is work in progress with Vladimir Pestov.