Largest components in random hypergraphs

TL;DR

This paper investigates the phase transition for the emergence of large $j$-tuple-connected components in random $k$-uniform hypergraphs, identifying the precise threshold and extending graph results to hypergraphs.

Contribution

It introduces a new bounded degree lemma and extends the depth-first search method to hypergraphs, providing a detailed analysis of component emergence.

Findings

Threshold for giant component emergence at edge probability rac{(k-j)!}{inom{k}{j}-1}n^{j-k}

Existence of a rac{(k-j)!}{inom{k}{j}-1}n^{j-k} phase transition

Extension of graph connectivity results to hypergraphs

Abstract

In this paper we consider $j$-tuple-connected components in random $k$-uniform hypergraphs (the $j$-tuple-connectedness relation can be defined by letting two $j$-sets be connected if they lie in a common edge and consider the transitive closure; the case $j=1$ corresponds to the common notion of vertex-connectedness). We determine that the existence of a $j$-tuple-connected component containing $\Theta (n^j)$ $j$-sets in random $k$-uniform hypergraphs undergoes a phase transition and show that the threshold occurs at edge probability $\tfrac{(k-j)!}{\binom{k}{j}-1}n^{j-k}$. Our proof extends the recent short proof for the graph case by Krivelevich and Sudakov which makes use of a depth-first search to reveal the edges of a random graph. Our main original contribution is a "bounded degree lemma" which controls the structure of the component grown in the search process.