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!



An upper bound on the families of large k-uniform sets with small pairwise intersection
Small pairwise intersection is eqhuivalent to large symmetric difference. So you are asking about constant weight error correcting codes. You can move to complenents and assume that k is at most n/2. There is large literature about it.


