Helly's theorem for systolic complexes

TL;DR

This paper establishes Helly's theorem analogues for systolic complexes, demonstrating bounds on Helly dimension for 6- and 7-systolic complexes, thus advancing understanding of their geometric properties.

Contribution

It introduces Helly's theorem analogues for systolic complexes and provides bounds on Helly dimension for 6- and 7-systolic complexes, a novel geometric insight.

Findings

7-systolic complexes have Helly dimension ≤ 1

6-systolic complexes have Helly dimension ≤ 2

Provides new bounds on Helly dimension for systolic complexes

Abstract

We prove the analogue of Helly's theorem for systolic complexes. Namely, we show that 7-systolic complexes have Helly dimension less or equal to 1, whereas 6-systolic complexes have Helly dimension bounded from the above by 2.