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A geometric graph is a graph G = (V; E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight line segments between points of V. Two edges of a geometric graph are said to be parallel, if they are opposite sides of a convex quadrilateral. In this paper we show that, for any xed k 3, any(More)
We prove a fractional version of the Erd˝ os–Szekeres theorem: for any k there is a constant c k > 0 such that any sufficiently large finite set X ⊂ R 2 contains k subsets Y 1 ,. .. , Y k , each of size ≥ c k |X |, such that every set {y 1 ,. .. , y k } with y i ∈ Y i is in convex position. The main tool is a lemma stating that any finite set X ⊂ R d(More)
Dedicated to our friend Imre Bárány on the occasion of his 50-th birthday. Abstract For any λ > 1 we construct a periodic and locally finite packing of the plane with ellipses whose λ-enlargement covers the whole plane. This answers a question of Imre Bárány. On the other hand, we show that if C is a packing in the plane with circular discs of radius at(More)