Paths, cycles and sprinkling in random hypergraphs

TL;DR

This paper establishes a lower bound on the length of the longest j-tight cycle in random hypergraphs, introducing a novel extension argument to connect paths into cycles beyond standard methods.

Contribution

It provides a new lower bound for cycle length in hypergraphs and develops an extension technique to connect paths into cycles, surpassing traditional sprinkling approaches.

Findings

Proves existence of long j-tight paths in random hypergraphs

Introduces an extension method to close paths into cycles

Establishes a lower bound on cycle length in hypergraphs

Abstract

We prove a lower bound on the length of the longest $j$-tight cycle in a $k$-uniform binomial random hypergraph for any $2 \le j \le k-1$. We first prove the existence of a $j$-tight path of the required length. The standard "sprinkling" argument is not enough to show that this path can be closed to a $j$-tight cycle -- we therefore show that the path has many extensions, which is sufficient to allow the sprinkling to close the cycle.