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A subset of a group is product-free if it does not contain elements a, b, csuch that ab = c. We review progress on the problem of determining the size ofthe largest product-free subset of an arbitrary finite group, including a lowerbound due to the author, and a recent upper bound due to Gowers. The bound ofGowers is more general; it allows three different sets A, B, C such that onecannot solve ab = c with a in A, b in B, c in C. We exhibit a refinement of thelower bound construction which shows that for this broader question, the boundof Gowers is essentially optimal.

Product-free subsets of groups, then and now