Let $X$ be a set with $n$ elements. I would like to know the maximum number of subsets of $X$ such that the number of elements in the symmetric difference between any two of these subsets is at most $k$.
For example if $k=n$, then the answer is $2^n$.
The maximum is attained at the family of all subsets of cardinality $\leq k/2$ if $k$ is even, and of all subsets whose intersection with $X\setminus\{x\}$ is of cardinatliy at most $(k-1)/2$ if $k$ is odd (here $x$ is some fixed element in $X$). This is proved by Kleitman in D. Kleitman, On a combinatorial conjecture of Erdös, J. Combin. Theory, Vol. 1, 209–214, at least for even values of $k$. P. Frankl attributes the full version to Kleitman (see Theorem 5.3 in P, Frankl, Extremal Set Systems, in R.L. Graham, M. Grötschel, L. Lovász (eds.), Handbook of Combinatorics, Vol. 2, Elsevier, 1995; pp. 1293--1330), but I am not sure whether Kleitman has actually claimed this. Anyway, the full version is proved in the cited Frankl's review.
