TITLE: Maximum intersecting subsets
QUESTION [8 upvotes]: There are n elements. What is the maximum number of subsets chosen at any one time so that every pair of subsets from collection intersect?

REPLY [2 votes]: As Sean Eberhard says, if $\mathcal{A}$ contains more than $2^{n−1}$ subsets of $[n]$ then it contains at least one pair of complementary subsets, so $2^{n−1}$ is an upper bound for $n\ge 1$.
It is easy to construct an example meeting this upper bound: simply let $\mathcal{B}\subset\mathcal{P}([n])$ contain just those subsets of $[n]$ which have $n$ as an element.  $\mathcal{B}$ clearly contains $2^{n-1}$ subsets of $[n]$ and the intersection of any positive number of them contains $n$.