Arcs on spheres intersecting at most twice

TL;DR

This paper determines the maximum number of arcs on a punctured sphere that intersect at most twice, providing exact formulas based on the number of punctures, which advances understanding of geometric intersection properties.

Contribution

The paper establishes exact maximum cardinalities for sets of arcs with limited intersections on punctured spheres, a novel result in geometric topology.

Findings

Maximum arcs intersecting at most once: |X|(|X| + 1)

Maximum arcs with arbitrary endpoints intersecting at most twice: |X|(|X| + 1)(|X| + 2)

Provides explicit formulas for intersection limits on punctured spheres.

Abstract

Let p be a puncture of a punctured sphere, and let Q be the set of all other punctures. We prove that the maximal cardinality of a set of arcs pairwise intersecting at most once, which start at p and end in Q, is |X|(|X| + 1). We deduce that the maximal cardinality of a set of arcs with arbitrary endpoints pairwise intersecting at most twice is |X|(|X| + 1)(|X| + 2).