The triangle and the open triangle

TL;DR

This paper proves that in percolation theory on transitive graphs, satisfying the triangle condition necessarily implies the open triangle condition, establishing a key theoretical link.

Contribution

It establishes a theoretical implication between the triangle condition and the open triangle condition in percolation on transitive graphs.

Findings

Triangle condition implies open triangle condition

Results apply to all transitive graphs

Strengthens understanding of percolation properties

Abstract

We show that for percolation on any transitive graph, the triangle condition implies the open triangle condition.