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Let A = {a 1 ,. .. , a k } and B = {b 1 ,. .. , b k } be two subsets of an Abelian group G, k ≤ |G|. Snevily conjectured that, when G is of odd order, there is a permutation π ∈ S k such that the sums a i + b π(i) , 1 ≤ i ≤ k, are pairwise different. Alon showed that the conjecture is true for groups of prime order, even when A is a sequence of k < |G|(More)
Let A and B be nonempty subsets of a finite group G in which the order of the smallest nonzero subgroup is not smaller than d = |A| + |B| − 1. Then at least d different elements of G has a representation in the form ab, where a ∈ A and b ∈ B. This extends a classical theorem of Cauchy and Davenport to noncommutative groups. We also generalize Vosper's(More)
In this expository paper we collect some combinatorial problems in the additive theory that can be easily solved in ordered Abelian groups. We study how such results, obtained by simple combinatorial arguments, can be extended to other Abelian groups. In many cases, best results can be obtained with the help of the so-called polynomial method that has(More)
Gyula ~iroly~ Institute for Advanced Study, Princeton Gbza T6th$ Courant Institute, Given a geometric graph, i.e., a collection of segments (edges) between n points in the plane, does it contain a non-crossing configuration of a certain type? It is widely conjectured that all such problems are NP–hard. This has been verified in many special cases, including(More)
Let A be a set of k elements of an Abelian group G in which the order of the smallest nonzero subgroup is at least 2k − 3. Then the number of different elements of G that can be written in the form a + a , where a, a ∈ A, a = a , is at least 2k − 3, as it has been shown in [12]. Here we give yet another proof of this result.