Monotone paths in random hypergraphs

TL;DR

This paper investigates the probability thresholds for the existence of monotone paths of various lengths in random oriented graphs derived from hypergraphs, providing insights into their structural properties.

Contribution

It determines the probability thresholds for monotone paths in random hypergraph-derived graphs, a novel analysis of their existence conditions.

Findings

Thresholds for finite monotone paths identified

Thresholds for infinite monotone paths established

Results applicable to line graphs of uniform hypergraphs

Abstract

We determine the probability thresholds for the existence of monotone paths, of finite and infinite length, in random oriented graphs with vertex set $\mathbb N^{[k]}$, the set of all increasing $k$-tuples in $\mathbb N$. These graphs appear as line graph of uniform hypergraphs with vertex set $\mathbb N$.