Quasirandom Groups

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.