How many sprays cover space?

Alessandro Andretta and Ivan Izmestiev

Vol. 341 (2026), No. 1, 1–32

DOI: 10.2140/pjm.2026.341.1

Abstract

For all $d\ge 3$ we show that the cardinality of $ℝ$ is at most ${\aleph }_n$ if and only if ${ℝ}^d$ can be covered with $(n+1)(d-1)+1$ sprays whose centers are in general position in a hyperplane. This extends previous results by Schmerl when $d=2$.

Keywords

continuum hypothesis, spray

Mathematical Subject Classification

Primary: 03E50

Secondary: 51N20

Milestones

Received: 11 October 2024

Revised: 26 August 2025

Accepted: 6 November 2025

Published: 10 March 2026