Forbidding Exactly One Hamming Distance

TL;DR

This paper determines the asymptotic size of the largest s-independent sets in the r-distance graph on the Boolean cube, using combinatorial and algebraic methods.

Contribution

It provides the first asymptotic characterization of the s-independence number for fixed even r in r-distance graphs, connecting extremal set theory and coding theory.

Findings

The s-independence number is Theta(2^n / n^{r/2}) for fixed s ≥ 2 and even r ≥ 2.

Upper bounds are obtained via sunflower-free set system extremal problems.

Lower bounds are constructed using BCH codes and constant-weight codes.

Abstract

Addressing questions raised in recent papers, we study the $r$-distance graph $H_r(n)$ on the Boolean cube $\{0,1\}^n$, where two vertices are adjacent if their Hamming distance is exactly $r$. For fixed integers $s \ge 2$ and even $r \ge 2$, we determine the asymptotic order of the $s$-independence number $\alpha_s(H_r(n))$, showing that \[ \alpha_s\left(H_r(n)\right)=\Theta\left(\frac{2^n}{n^{r/2}}\right). \] The upper bound is derived via a reduction to extremal problems for sunflower-free set systems, while the lower bound is obtained using algebraic constructions based on BCH codes and constant-weight codes.