The Dirac--Goodman--Pollack Conjecture

TL;DR

This paper confirms the Dirac--Goodman--Pollack conjecture in allowable sequences, establishing a specific bound and advancing understanding of geometric configurations through combinatorial methods.

Contribution

The paper proves the Dirac--Goodman--Pollack conjecture with a concrete bound, providing new insights into allowable sequences and their geometric implications.

Findings

Confirmed the conjecture with bound c=1/845

Analyzed properties of allowable sequences

Discussed related geometric problems

Abstract

In one of their seminal articles on allowable sequences, Goodman and Pollack gave combinatorial generalizations for three problems in discrete geometry, one of which being the Dirac conjecture. According to this conjecture, any set of $n$ noncollinear points in the plane has a point incident to at least $c n$ connecting lines determined by the set. The notion of allowable sequences of permutations provides a natural combinatorial setting for analyzing these problems. Within this formalism, the conjectured generalization reads as follows: \emph{Any nontrivial allowable $n$-sequence $\Sigma$ has a local sequence $\Lambda_i$ whose half-period is at least $c n$.} The conjecture is confirmed here with a concrete bound $c=1/845$. Several related problems are discussed.