Corpus ID: 16437438

Observations and Computations in Sylvester-Gallai Theory

  title={Observations and Computations in Sylvester-Gallai Theory},
  author={Jonathan Lenchner and Herv{\'e} Br{\"o}nnimann},
We bring together several new results related to the classical Sylvester-Gallai Theorem and its recently noted sharp dual. In 1951 Dirac and Motzkin conjectured that a configuration of n not all collinear points must admit at least n/2 ordinary connecting lines. There are two known counterexamples, when n = 7 and n = 13. We provide a construction that yields both counterexamples and show that the common construction cannot be extended to provide additional counterexamples. 


A survey of Sylvester's problem and its generalizations
SummaryLetP be a finite set of three or more noncollinear points in the plane. A line which contains two or more points ofP is called aconnecting line (determined byP), and we call a connectingExpand
Sylvester's Problem on Collinear Points
(1968). Sylvester's Problem on Collinear Points. Mathematics Magazine: Vol. 41, No. 1, pp. 30-34.
There exist 6n/13 ordinary points
One of the main theorems used by Hansen is false, thus leavingn/2 open, and the 3n/7 estimate is improved to 6n/13 (apart from pencils and the Kelly-Moser example), with equality in the McKee example. Expand
Geometric Graphs and Arrangements
Preface The body of this text is written. It remains to find some words to explain what to expect in this book. A first attempt of characterizing the content could be: In words: The questions posedExpand
Wedges in Euclidean Arrangements
Given an arrangement of n not all coincident lines in the Euclidean plane we show that there can be no more than $\lfloor 4n/3\rfloor$ wedges (i.e. two-edged faces) and give explicit examples to showExpand
The lines and planes connecting the points of a finite set
0.1. Not later than 1933 I made the following conjecture, originally in the form of a statement on the minors of a matrix(1). T<¡. Any n points in d-space that are not on one hyper plane determine atExpand
On the dual and sharpened dual of Sylvester's theorem in the plane
  • IBM Research Report
  • 2004
On the number of ordinary lines determined by $n$ points