Tahun Terbit

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The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Let P ⊂ (0, 1] be the set of commuting probabilities of all finite groups. We prove that every point of P is nearly an Egyptian fraction of bounded complexity. As a corollary, we deduce two conjectures of Keith Joseph from 1977: all limit points of P are rational, …

Erd˝os showed that every set of n positive integers contains a subset of size at least n/(k + 1) containing no solutions to x1 + · · · + xk = y. We prove that the constant 1/(k + 1) here is best possible by showing that if (Fm) is a multiplicative Følner sequence in N, then Fm has no k-sum-free subset of size greater than (1/(k + 1) + o(1))|Fm|.