Existence thresholds and Ramsey properties of random posets

TL;DR

This paper investigates the thresholds for induced subposet containment, counts copies of fixed posets in the Boolean lattice, and explores Ramsey properties of random posets generated by probabilistic inclusion.

Contribution

It provides the first precise thresholds for induced subposet containment and analyzes the Ramsey properties of random posets, extending classical combinatorial results.

Findings

Determined the threshold for containing a fixed poset as an induced subposet.

Asymptotically counted the number of copies of a fixed poset in the Boolean lattice.

Established new results on the Ramsey properties of random posets.

Abstract

Let $\mathcal P(n)$ denote the power set of $[n]$, ordered by inclusion, and let $\mathcal P (n,p)$ denote the random poset obtained from $\mathcal P(n)$ by retaining each element from $\mathcal P (n)$ independently at random with probability $p$ and discarding it otherwise. Given any fixed poset $F$ we determine the threshold for the property that $\mathcal P(n,p)$ contains $F$ as an induced subposet. We also asymptotically determine the number of copies of a fixed poset $F$ in $\mathcal P(n)$. Finally, we obtain a number of results on the Ramsey properties of the random poset $\mathcal P(n,p)$.