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Two Proofs for Sylvester's Problem Using an Allowable Sequence of Permutations

Published online by Cambridge University Press:  27 June 2025

Jacob E. Goodman
Affiliation:
City College, City University of New York
Janos Pach
Affiliation:
City College, City University of New York and New York University
Emo Welzl
Affiliation:
Eidgenössische Technische Hochschule Zürich
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Summary

The famous Sylvester's problem is: Given finitely many noncollinear points in the plane, do they always span a line that contains precisely two of the points? The answer is yes, as was first shown by Gallai in 1944. Since then, many other proofs and generalizations of the problem appeared. We present two new proofs of Gallai's result, using the powerful method of allowable sequences.

1. Introduction

Sylvester [1893] raised the following problem: Given finitely many noncollinear points in the plane, do they always span a simple line (that is, a line that contains precisely two of the points)? The answer is yes, as was first shown by Gallai [1944].

By duality, the former question is equivalent to the question: Given finitely many straight lines in the plane, not all passing through the same point, do they always determine a simple intersection point (a point that lies on precisely two of the lines)?

A natural generalization is to find a lower bound on the number of simple lines (or simple points, in the dual version). The dual version of this question can be generalized to pseudolines. The best lower bound [Csima and Sawyer 1993] states that an arrangement of n pseudolines in the plane determines at least 6n/13 simple points. The conjecture [Borwein and Moser 1990] is that there are at least n/2 simple points for n ≠ 7,13. For the history of Sylvester's problem, with its many proofs and generalizations, see [Borwein and Moser 1990; Nilakantan 2005].

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Publisher: Cambridge University Press
Print publication year: 2005

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