On partial Steiner $(n,r,\ell)$-system process

TL;DR

This paper analyzes the connectivity threshold in a process of building partial Steiner hypergraphs, showing it occurs sharply at a specific edge count regardless of the parameter , with the last edge linking the final isolated vertex.

Contribution

It establishes the sharp connectivity threshold for the partial Steiner -system process, independent of , and characterizes the critical edge that ensures connectivity.

Findings

Connectivity threshold is  rac{n}{r} ext{log} n.

The last edge connecting the final isolated vertex determines connectivity.

Threshold is sharp and independent of .

Abstract

For given integers $r$ and $\ell$ such that $2\leqslant\ell\leqslant r-1$, an $r$-uniform hypergraph $H$ is called a partial Steiner $(n,r,\ell)$-system, if every subset of size $\ell$ lies in at most one edge of $H$. In particular, partial Steiner $(n,r,2)$-systems are also called linear hypergraphs. The partial Steiner $(n,r,\ell)$-system process starts with an empty hypergraph on vertex set $[n]$ at time $0$, the $ \binom{n}{r}$ edges arrive one by one according to a uniformly chosen permutation, and each edge is added if and only if it does not overlap any of the previously-added edges in $\ell$ or more vertices. In this paper, we show with high probability, independent of $\ell$, the sharp threshold of connectivity in the algorithm is $ \frac{n}{r}\log n$ and the very edge which links the last isolated vertex with another vertex makes the partial Steiner $(n,r,\ell)$-system connected.