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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. It is known that, for any xed k, any geometric graph G on n vertices with no k pairwise crossing edges contains at most O(n log n) edges. In this paper we give a… (More)

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)

The extremal function Ex(u, n) (introduced in the theory of Davenport-Schinzel sequences in other notation) denotes for a fixed finite alternating sequence u = ababa. .. the maximum length of a finite sequence v over n symbols with no immediate repetition which does not contain u. Here (following the idea of J. Nešetřil) we generalize this concept for… (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)

Let g(n) denote the least integer such that among any g(n) points in general position in the plane there are always n in convex position. In 1935 P. Erd} os and G. Szekeres showed that g(n) exists and 2 n?2 + 1 g(n) 2n?4 n?2 + 1. Recently, the upper bound has been slightly improved by Chung and Graham and by Kleitman and Pachter. In this note we further… (More)

We analyze a randomized pivoting process involving one line and <italic>n</italic> points in the plane. The process models the behavior of the <italic>Random-Edge</italic> simplex algorithm on simple polytopes with <italic>n</italic> facets in dimension <italic>n-2</italic>. We obtain a tight <italic>O(\log^2 n)</italic> bound for the expected number of… (More)

Bob cuts a pizza into slices of not necessarily equal size and shares it with Alice by alternately taking turns. One slice is taken in each turn. The first turn is Alice's. She may choose any of the slices. In all other turns only those slices can be chosen that have a neighbor slice already eaten. We prove a conjecture of Peter Winkler by showing that… (More)