I am new here and I hope I don't make any mistakes in asking this question here.

Let $n > k$ be integers, and $\epsilon < 1/2$ be an arbitrarily small constant. What is an upper bound on the maximum size of a family $\{A_1,\ldots,A_N\}$ of subsets of $[n]$ such that:

- $|A_i| = k$
- $|A_i \cap A_j| \le (1-\epsilon)k$.

I know there are a bunch of results on this question but I am interested in getting good bounds in the regime where $k \ge n/2$, and I did not manage to get anything.

The most related result I have found is a lemma by Corradi that shows that, in the regime I am interested in, if $\epsilon \ge 1/2$, then the families can have size at most some function of $\epsilon$ (i.e.: constant independent of n). I am trying to find out what we can say when $\epsilon$ is a constant less than 1/2 and, importantly whether a constant bound applies (any loose constant bound will be great). If there is a way of seeing that the bound is not going to be constant, it would also be great.

Any direction would be highly appreciated. Thanks!