Set families: restricted distances via restricted intersections
Zichao Dong, Jun Gao, Hong Liu, Minghui Ouyang, Qiang Zhou
TL;DR
The paper resolves asymptotics for the maximal size of set families with restricted distances when the distance set $D$ is a homogeneous arithmetic progression, linking restricted-distance problems to restricted-intersection frameworks. It extends Kleitman’s isodiametric inequality in the discrete setting, confirms a conjecture for a broad class of $D$, and reveals a linear growth dichotomy for non-homogeneous $D$. The authors develop lower bounds via Rödl nibble constructions and design theory, and matching upper bounds through a synthesis of intersection theorems (Frankl–Wilson, Deza–Erdős–Frankl) and modular methods, with a final spectral argument to pinpoint the unique maximizer $D=igl\{2,4,\dots,2tigrigr ron}$ and the exact $t$-code bounds in Hamming cubes. The work also highlights applications to binary $t$-codes and suggests avenues for strengthening known bounds and open questions in non-homogeneous regimes.
Abstract
Denote by $f_D(n)$ the maximum size of a set family $\mathcal{F}$ on $[n] \stackrel{\mbox{\normalfont\tiny def}}{=} \{1, \dots, n\}$ with distance set $D$. That is, $|A \bigtriangleup B| \in D$ holds for every pair of distinct sets $A, B \in \mathcal{F}$. Kleitman's celebrated discrete isodiametric inequality states that $f_D(n)$ is maximized at Hamming balls of radius $d/2$ when $D = \{1, \dots, d\}$. We study the generalization where $D$ is a set of arithmetic progression and determine $f_D(n)$ asymptotically for all homogeneous $D$. In the special case when $D$ is an interval, our result confirms a conjecture of Huang, Klurman, and Pohoata. Moreover, we demonstrate a dichotomy in the growth of $f_D(n)$, showing linear growth in $n$ when $D$ is a non-homogeneous arithmetic progression. Different from previous combinatorial and spectral approaches, we deduce our results by converting the restricted distance problems to restricted intersection problems. Our proof ideas can be adapted to prove upper bounds on $t$-distance sets in Hamming cubes (also known as binary $t$-codes), which has been extensively studied by algebraic combinatorialists community, improving previous bounds from polynomial methods and optimization approaches.