I'm trying to come up with the largest family of sets that obeys the following properties:

Consider $X = \{1,\dots,n\}$ and take $\mathcal{F} \subset 2^X$ such that for any three subsets $A,B,C \in \mathcal{F}$ we have that (at least) two of the numbers $|A \cap B|, |B \cap C|, |A \cap C|$ are the same size.

There is a trivial example here, $\{1\}, \{1,2\}, \{1,2,3\},\dots$ which gives a family $\mathcal{F}$ of size $n$, but I don't see how to do better (I've tried using modular arithmetic, which doesn't seem to help all that much.)

In this problem it doesn't particularly matter (to me at least) what $n$ is, eg, for the modular arithmetic way I was trying I set $n = L^2$ for some integer.

So, my question is as follows, is there a better choice for $\mathcal{F}$, eg, one that is larger than the trivial?