Simple wedge points

TL;DR

This paper proves that under specific conditions, every simple line in a finite point set creates a simple wedge, and introduces a method to explore whether such wedges exist without those restrictions.

Contribution

The paper establishes the existence of simple wedges in certain finite point sets and introduces a new method for analyzing their presence.

Findings

Every simple line in V creates a simple wedge under given conditions.

Both the odd size of V and the limit on collinear points are necessary for the main result.

A new method is proposed for proving the existence of simple wedges in broader cases.

Abstract

Let V be a finite set of points in the plane, not contained in a line. Assume |V| = n is an odd number, and |L \cap V| \leq 3 for every line L which is spanned by V. We prove that every simple line L_{a,b} in V creates a simple wedge (i.e., a triple {a, b, c} \subseteq V such that L_{a,b} and L_{a,c} are simple lines). We also show that both restrictions on V (namely |V| is odd and |L \cap V| \leq 3) are needed. We conjecture, further, that if |V | = n is an odd number then V contains a simple wedge, even if V is not 3-bounded. We introduce a method for proving this, which gives (in this paper) partial results.