The following question is a open question related to coding theory : What is the maximal size of a collection of $(\frac{n}{2} + 1)$-elements subsets of an n-element set such that each pair of subsets has at most $\frac{n}{2}-1$ elements in common ? We just have lower bound which is : $(\frac{1}{n} + O(\frac{1}{n^2})){n \choose {n/2}}$.


Now, instead of uniform subset, if we consider a collection of subsets with at least $\frac{n}{2} + 1$ elements such that each pair of subsets has at most $\frac{n}{2}-1$ elements in common, do we have a better lower bound for the size of the collection ?



Lower bound of the size of a collection of subsets with a intersecting property
The two problems are equivalent. Indeed, let $\mathcal{F}$ be the family of sets of size at least $n/2+1$ such that no two sets have $n/2$ elements in common. Clearly, such an $\mathcal{F}$ cannot contain two sets $S,T$ such that $S\subset T$. If there is a set $S\in \mathcal{F}$ of size greater than $n/2+1$, then replace it by a set $S'=S\setminus\{x\}$ where $x$ is any element of $S$. The result is the family $\mathcal{F}'=\mathcal{F}\cup \{S'\}\setminus \{S\}$ which has the same size as $\mathcal{F}$, and satisfies the same condition. Repeat as long as there are sets of size greater than $n/2+1$.


