The random forbidden-subposet threshold conjecture

Let $P$ be a finite connected poset. Let $e(P)$ and $e^*(P)$ be the consecutive-level parameters, and let $d(P)$ and $d^*(P)$ be the minimum ratios defined over connected subposets having the same corresponding $e$-parameter. Write $\mathcal P(n,p)$ for the random subfamily of $2^{[n]}$ obtained by retaining each set independently with probability $p$, and let $\omega$ denote a quantity tending to infinity. The random threshold conjecture. If $p=\omega(n^{-d(P)})$, then the largest $P$-free family in $\mathcal P(n,p)$ has size

$$(e(P)+o(1))p\binom{n}{\lfloor n/2\rfloor}$$

with high probability; if $p=\omega(n^{-d^*(P)})$, then the largest induced $P$-free family has size

$$(e^*(P)+o(1))p\binom{n}{\lfloor n/2\rfloor}$$

with high probability.

This conjecture proposes sharpness of the preceding lower bounds in the random setting. The source supplies no general proof or disproof.