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- Pavel Valtr
- Graph Drawing
- 1997

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)

- Martin Klazar, Pavel Valtr
- Combinatorica
- 1994

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)

- Pavel Valtr
- Discrete & Computational Geometry
- 1998

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)

- Imre Bárány, Pavel Valtr
- Discrete & Computational Geometry
- 1998

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)

- Radek Adamec, Martin Klazar, Pavel Valtr
- Discrete Mathematics
- 1992

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)

- Krystyna Trybulec Kuperberg, Wlodzimierz Kuperberg, Jirí Matousek, Pavel Valtr
- Discrete & Computational Geometry
- 1999

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)

- 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)

- Géza Tóth, Pavel Valtr
- Discrete & Computational Geometry
- 1998

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)