Forbidden patterns and unit distances

János Pach, G. Tardos

Research output: Chapter in Book/Report/Conference proceedingConference contribution

5 Citations (Scopus)

Abstract

At most how many edges (hyperedges, nonzero entries, characters) can a graph (hypergraph, zero-one matrix, string) have if it does not contain a fixed forbidden pattern? Turántype extremal graph theory, Erdos-Ko-Rado-type extremal set theory, Ramsey theory, the theory of Davenport-Schinzel sequences, etc. have been developed to address questions of this kind. They produced a number of results that found important applications in discrete and computational geometry. In the present paper, we discuss an extension of extremal graph theory to ordered graphs, i.e., to graphs whose vertex set is linearly ordered. In the most interesting cases, the for-bidden ordered graphs are bipartite, and the basic problem can be reformulated as an extremal problem for zero-one matrices avoiding a certain submatrix P. We disprove a general conjecture of Füredi and Hajnal related to the latter problem, and replace it by some weaker alternatives. We verify our conjectures in a few special cases when P is the adjacency matrix of an acyclic graph and discuss the same question when the forbidden patterns are adjacency matrices of cycles. Our results lead to a new proof of the celebrated theorem of Spencer, Szemerédi, and Trotter [15] stating that the number of times that the unit distance can occur among n points in the plane is O(n4/3). This is the first proof that does not use any tool other than a forbidden pattern argument. We present another geometric application, where the forbidden pattern P is the adjacency matrix of an acyclic graph. A hippodrome is a c × d rectangle with two semidisks of diameter d attached to its sides of length d. Improving a result of Efrat and Sharir [5], we show that the number of "free" placements of a convex n-gon in general position in a hippodrome H such that simultaneously three vertices of the polygon lie on the boundary of H, is O(n). This result is related to the Planar Segment-Center Problem.

Original languageEnglish
Title of host publicationProceedings of the Annual Symposium on Computational Geometry
Pages1-9
Number of pages9
DOIs
Publication statusPublished - 2005
Event21st Annual Symposium on Computational Geometry, SCG'05 - Pisa, Italy
Duration: Jun 6 2005Jun 8 2005

Other

Other21st Annual Symposium on Computational Geometry, SCG'05
CountryItaly
CityPisa
Period6/6/056/8/05

Fingerprint

Unit
Adjacency Matrix
Graph in graph theory
Extremal Graph Theory
Graph theory
Extremal Set Theory
Computational geometry
Ramsey Theory
Discrete Geometry
n-gon
Center Problem
Set theory
Disprove
Extremal Problems
Computational Geometry
Zero
Erdös
Hypergraph
Rectangle
Placement

Keywords

  • 0-1 matrix
  • Extremal combinatorics
  • Forbidden patterns
  • Interval chromatic number
  • Unit-distance graph

ASJC Scopus subject areas

  • Software
  • Geometry and Topology
  • Safety, Risk, Reliability and Quality
  • Chemical Health and Safety

Cite this

Pach, J., & Tardos, G. (2005). Forbidden patterns and unit distances. In Proceedings of the Annual Symposium on Computational Geometry (pp. 1-9) https://doi.org/10.1145/1064092.1064096

Forbidden patterns and unit distances. / Pach, János; Tardos, G.

Proceedings of the Annual Symposium on Computational Geometry. 2005. p. 1-9.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Pach, J & Tardos, G 2005, Forbidden patterns and unit distances. in Proceedings of the Annual Symposium on Computational Geometry. pp. 1-9, 21st Annual Symposium on Computational Geometry, SCG'05, Pisa, Italy, 6/6/05. https://doi.org/10.1145/1064092.1064096
Pach J, Tardos G. Forbidden patterns and unit distances. In Proceedings of the Annual Symposium on Computational Geometry. 2005. p. 1-9 https://doi.org/10.1145/1064092.1064096
Pach, János ; Tardos, G. / Forbidden patterns and unit distances. Proceedings of the Annual Symposium on Computational Geometry. 2005. pp. 1-9
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