Kneser's theorem for codes and $\ell$-divisible set families

TL;DR

This paper extends Kneser’s theorem to a coding-theory framework to bound the size of $k$-wise $\ell$-divisible set families of subsets of $[n]$, marrying additive combinatorics with coding theory. The central technique embeds the family into a vector space over $\mathbb{F}_p$ and leverages a Hadamard product perspective along with Kneser’s Theorem for codes to control stabiliser growth through the sequence $V, V^{\langle 2\rangle}, \dots, V^{\langle k\rangle}$. For prime $\ell = p$, the authors prove tight bounds $|\mathcal{F}| \le 2^{\lfloor n/p\rfloor}$ with atomic extremals, and they extend the theory to composite $\ell$ showing $4\ell^2$-wise $\ell$-divisible families satisfy the same bound and are contained in atomic families with atoms of size $\ell$, improving on prior exponential-parameter requirements. These results advance Eventown-type bounds and illuminate the role of stabilisers and atoms in the extremal structure of set families, while connecting to recent work by Gishboliner, Sudakov, and Timon on weakened divisibility hypotheses.

Abstract

A $k$-wise $\ell$-divisible set family is a collection $\mathcal{F}$ of subsets of ${ \{1,\ldots,n \} }$ such that any intersection of $k$ sets in $\mathcal{F}$ has cardinality divisible by $\ell$. If $k=\ell=2$, it is well-known that $|\mathcal{F}|\leq 2^{\lfloor n/2 \rfloor}$. We generalise this by proving that $|\mathcal{F}|\leq 2^{\lfloor n/p\rfloor}$ if $k=\ell=p$, for any prime number $p$. For arbitrary values of $\ell$, we prove that $4\ell^2$-wise $\ell$-divisible set families $\mathcal{F}$ satisfy $|\mathcal{F}|\leq 2^{\lfloor n/\ell\rfloor}$ and that the only families achieving the upper bound are atomic, meaning that they consist of all the unions of disjoint subsets of size $\ell$. This improves upon a recent result by Gishboliner, Sudakov and Timon, that arrived at the same conclusion for $k$-wise $\ell$-divisible families, with values of $k$ that behave exponentially in $\ell$. Our techniques rely heavily upon a coding-theory analogue of Kneser's Theorem from additive combinatorics.

Figures (1)

• Figure 1: The four vectors of claim 3

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Theorem 9, Gishboliner2022Small
• Theorem 1.4: structure theorem for $\ell$-divisible set families
• Theorem 1.5: Theorem 1, Gishboliner2022Small
• Definition 2.1: stabiliser
• Proposition 2.2: Lemma 2.10, KneserCode
• Theorem 2.3: Theorem 3.3, KneserCode
• Lemma 2.4
• proof