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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)

- GYULA KÁROLYI
- 2005

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)

- GYULA KÁROLYI
- 2005

Let A be a set of k ≥ 5 elements of an Abelian group G in which the order of the smallest nonzero subgroup is larger than 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 [21]. Here we prove that the bound is attained if and only if the elements of… (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)

Let S be a point set in the plane in general position, such that its elements are partitioned into k classes or colors. In this paper we study several variants on problems related to the Erd˝ os-Szekeres theorem about subsets of S in convex position, when additional chromatic constraints are considered.

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 S be a point set in the plane in general position, such that its elements are partitioned into k classes or colors. In this paper we study several variants on problems related to the Erd˝ os-Szekeres theorem about subsets of S in convex position, when additional chromatic constraints are considered.

- GYULA KÁROLYI
- 2004

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.

Two triangles are called almost disjoint if they are either disjoint or their intersection consists of one common vertex. Let f (n) denote the maximum number of pairwise almost disjoint triangles that can be found on some vertex set of n points in 3-space. Here we prove that f (n) = (n 3/2).