On the intersecting family process

TL;DR

This paper investigates the intersecting family process for random sequences of k-sets, providing new theoretical results for specific growth regimes of k relative to n.

Contribution

It introduces new findings for the intersecting family process when k scales as n^{1/3} and when k grows much faster than n^{1/2}.

Findings

Results for k=cn^{1/3} case.

Results for k  n^{1/2} case.

Advances understanding of intersecting families in random set processes.

Abstract

We study the intersecting family process initially studied in \cite{BCFMR}. Here $k=k(n)$ and $E_1,E_2,\ldots,E_m$ is a random sequence of $k$-sets from $\binom{[n]}{k}$ where $E_{r+1}$ is uniformly chosen from those $k$-sets that are not already chosen and that meet $E_i,i=1,2,\ldots,r$. We prove some new results for the case where $k=cn^{1/3}$ and for the case where $k\gg n^{1/2}$.