How many $2k$-subsets of the integers $\{1, \dots, n\}$ have pairwise symmetric difference at least $k$?
Problem statement: Let $\binom{[n]}{2k}$ denote the family of $2k$-subsets on $[n]$. Say a sub collection $\mathcal{F} \subset \binom{[n]}{2k}$ is $k$-separated if $$ S \neq T \in \mathcal{F} \implies |S \Delta T| \geq k. $$ I am interested in the cardinality (or bounds on the cardinality) of the largest family $\mathcal{F}$: $$ M_n(k) := \max\Big\{\mathcal{F} \subset \binom{[n]}{2k} : \mathcal{F}~\mbox{is $k$-separated}\Big\} $$
I know that each $2k$-subset must exclude $\sum_{j=0}^{k-1} \binom{n-2k}{j} \binom{2k}{2k - j}$ other subsets, simply by considering where the overlaps can be. However, I don't know how to relate this to the overall number of subsets.