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 ?