For $X=\lbrace 0,\ldots,n-1\rbrace$, let $F\subseteq 2^X$ be a family of subsets of $X$ such that, for every $x\in X$, the singleton $\lbrace x\rbrace$ is the intersection of some elements of $F$. I am interested in the minimal families that have this property, in particular whether it is possible to have $|F|< n$. Can anyone (a) give an example where $|F|< n$, (b) provide an argument for why $|F|\ge n$, or (c) point me in the direction of some existing results.


Bonus question: For any $k< n$, the family $F= \lbrace \lbrace x,x+1,\ldots,x+k-1 \rbrace:x\in X \rbrace$, where addition is carried out modulo $n$, satisfies the stated condition (minimally) and contains exactly $n$ elements, so $|F|= n$ is always achievable. How does the structure of a general minimal family relate to these highly regular families? Is a minimal family always a disjoint union of some regular families?



Families of subsets containing every singleton as an intersection
You can achieve $\lvert F\rvert = 2\lceil\log\_2 n\rceil$ by using all subsets of the form $\{x\in X \vert i^{\text{th}}\text{ bit of }x\text{ is }j\}$ for $i \in \{0,1,\ldots,\lceil\log\_2 n\rceil-1\}$ and $j\in\{0,1\}$.


This rate is within a factor of two of best possible because there are at most $2^{\lvert F\rvert}$ different intersections you can form from $\lvert F\rvert$ sets. You require that all $n$ singletons be among this list of intersections, so $\lvert F\rvert \geq \log\_2 n$.


Are you interested in the actual minimal value of $\lvert F\rvert$, or just asymptotics?


