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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 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)