TITLE: Extensions of Sperner families on finite sets
QUESTION [8 upvotes]: Let $V$ be a set consisting of $n$ points, $n\geq 2$.
A Sperner family on $V$ is a set $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$, $m\geq 1$, where each $\sigma_i$, $i=1,2,\dots,m $, is a nonempty subset of $V$, and for any $i\neq j$, $\sigma_i$ does not contain $\sigma_j$.
Given a Sperner family $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$, an extension of it is a new Sperner family $\{\sigma_1,\sigma_2,\dots,\sigma_m,\sigma_{m+1}\}$ obtained by adding a subset $\sigma_{m+1}$ of $V$ to $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$.
Question: Let $R(\sigma_1,\sigma_2,\dots,\sigma_m)$ be the number of extensions of the Sperner family $\{\sigma_1,\sigma_2,\dots,\sigma_m\}$. Are there any methods or formulas to compute $R(\sigma_1,\sigma_2,\dots,\sigma_m)$? Are there any references?
Thanks.

REPLY [4 votes]: If your concern is computational complexity then one can
avoid the term $2^m$ which tends to be doubly exponential in $n$.
One can solve this problem in time $\text{O}(mn 2^n)$ simply by checking
for each subset of $V$ whether or not it is comparable to some
element of your Sperner family. (Here the factor $n$ accounts
for the cost of such a comparison.)
I do not think that there is a much faster algorithm as $\text{O}(mn 2^n)$, I
mean, all known algorithms are exponential. Because it should read at least a matrix of size $n \times m$.
CW Complex, Einar, and others, if you are further interested in the area, I can suggest you a related problem to solve which is (probably) more difficult.
Find the maximum of the number $R(\sigma_1, \ldots)$ when the number of sets, $m$ is fixed (but the sets can be moved). Start with the large values of $m$. Let $M$ denote the largest possible value of $m$: the results of the Sperner theorem ($=$ the largest binomial coefficient of order $n$). The problem is trivial for$$m=M, M-1, M-2, \ldots.$$It starts to be less trivial around $M-n/2\ldots$.