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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.
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Gowers, W. T. (2008). Quasirandom Groups. Combinatorics Probability Computing, 17(3), 363–387. https://doi.org/10.1017/s0963548307008826
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