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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)

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)

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)

A subset A of a finite set P of points in the plane is called an empty polygon, if each point of A is a vertex of the convex hull of A and the convex hull of A contains no other points of P. We construct a set of n points in general position in the plane with only ≈ 1.62n 2 empty triangles, ≈ 1.94n 2 empty quadrilaterals, ≈ 1.02n 2 empty pentagons, and ≈… (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)

- PAVEL VALTR
- 2005

Let ES(n) denote the least integer such that among any ES(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 ES(n) exists and ES(n) ≤ ` 2n−4 n−2´+ 1. Six decades later, the upper bound was slightly improved by Chung and Graham, a few months later it was further improved by… (More)