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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 ~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.

We show that for any 2{coloring of the ? n 2 segments determined by n points in the plane, one of the color classes contains non-crossing cycles of lengths 3; 4; : : : ; b p n=2c. This result is tight up to a multi-plicative constant. Under the same assumptions, we also prove that there is a non-crossing path of length (n 2=3), all of whose edges are of the… (More)

Let C n denote the cycle of length n. The generalized Ramsey number of the pair (C n ; C k), denoted by R(C n ; C k), is the smallest positive integer R such that any complete graph with R vertices whose edges are coloured with two diierent colours contains either a monochromatic cycle of length n in the rst colour or a monochromatic cycle of length k in… (More)

In certain families of hypergraphs the transversal number is bounded by some function of the packing number. In this paper we study hypergraphs related to multiple intervals and axis-parallel rectangles, respectively. Essential improvements of former established upper bounds are presented here. We explore the close connection between the two problems at… (More)