# On subsets of the hypercube with prescribed Hamming distances

## TL;DR

This paper provides an algebraic proof of Kleitman's theorem on the maximum size of binary vector sets with a given Hamming diameter, extending it to other distance sets and improving related bounds.

## Contribution

It introduces an algebraic approach using pseudo-adjacency matrices to prove and extend Kleitman's theorem and related results in combinatorics.

## Key findings

- Algebraic proof of Kleitman's theorem using Cvetković bound
- Extensions to other distance sets beyond linear growth
- Improved bounds on subsets of finite fields with restricted difference sets

## Abstract

A celebrated theorem of Kleitman in extremal combinatorics states that a collection of binary vectors in $\{0, 1\}^n$ with diameter $d$ has cardinality at most that of a Hamming ball of radius $d/2$. In this paper, we give an algebraic proof of Kleitman's Theorem, by carefully choosing a pseudo-adjacency matrix for certain Hamming graphs, and applying the Cvetkovi\'c bound on independence numbers. This method also allows us to prove several extensions and generalizations of Kleitman's Theorem to other allowed distance sets, in particular blocks of consecutive integers that do not necessarily grow linearly with $n$. We also improve on a theorem of Alon about subsets of $\mathbb{F}_{p}^{n}$ whose difference set does not intersect $\left\{0,1\right\}^{n}$ nontrivially.