# Union closed family of sets with at most a certain number of couples of sets with non-empty intersection

Is it possible to find a union closed family $$\mathcal{F}$$, $$\emptyset \notin \mathcal{F}$$, with $$|\mathcal{F}| = n$$ sets, such that there are at most:

$$\left(1-\frac{1}{\left\lfloor \frac{n-1}{2} \right\rfloor}\right)\frac{n^2}{2}$$

unordered couples of sets in it with at least one element in common?

If that is not possible, and if we build a graph such that every vertex represents a set and any edge exists if and only if the two corresponding sets have an element in common, by Turán's theorem there will be a $$K_{\lceil \frac{n}{2} \rceil}$$ complete subgraph in it, and therefore a subfamily of $$\mathcal{F}$$ of size $$\lceil \frac{n}{2} \rceil$$ where each unordered couple of sets has an element in common. Maybe from that subfamily it will be then possible to build another subfamily of size $$\lceil \frac{n}{2} \rceil$$ with the same element in common among all sets.

Note that every union closed family with $$n$$ sets and without the empty set has at least $$\frac{2}{3}\binom{n}{2}$$ unordered couples of sets with at least one element in common, because for every couple without an element in common $$\{A,B\}$$ we can build two unique other couples $$\{A,A \cup B\}$$ and $$\{B,A \cup B\}$$ with non-empty intersection. However $$\frac{2}{3}\binom{n}{2}$$ is bigger than the above value only for $$n \lt 7$$.

Since a family as required by the question would imply that there are $$\frac{n^2}{2\lfloor \frac{n-1}{2} \rfloor} \gt n$$ unordered couples of sets with empty intersection, maybe it is possible to show that these couples would generate more than $$n$$ sets, or that to build those couples we need more than the maximum number of elements ($$\frac{n-1}{4}$$) required for a possible counterexample of the union closed sets conjecture.

Well, thinking more carefully I have found the simple counterexample given by the powerset on $$n$$ elements without the empty set, $$\mathcal{P}([n]) \setminus \emptyset$$, for $$n \ge 4$$.