a1 Department of Pure Mathematics and Mathematical Statistics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK (e-mail: email@example.com)
Babai and Sós have asked whether there exists a constant c > 0 such that every finite group G has a product-free subset of size at least c|G|: that is, a subset X that does not contain three elements x, y and z with xy = z. In this paper we show that the answer is no. Moreover, we give a simple sufficient condition for a group not to have any large product-free subset.
(Received July 05 2006)
(Revised October 27 2007)