A subset of a group is said to be product free if it does not contain three elements satisfying the equation $xy=z$. We give a negative answer to a question of Babai and Sós on the existence of large product‐free sets in finite groups by model theoretic means. This question was originally answered by Gowers. Furthermore, we give a natural and sufficient model theoretic condition for a group to have a large product‐free subset, as well as a model theoretic account of a result of Nikolov and Pyber on triple products.

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关于压实和无产品组合

如果一个组的子集不包含满足等式的三个元素，则称该子集为无乘积 $Xÿ=ž$。对于Babai和Sós关于通过模型理论方法在有限组中存在无乘积集的问题，我们给出否定的答案。这个问题最初是由高尔斯回答的。此外，我们为一个具有大量无产品子集的组提供了自然而充分的模型理论条件，并给出了Nikolov和Pyber对三重产品的结果的模型理论说明。