Sperner systems with restricted differences

TL;DR

This paper establishes new upper bounds on $q$-modular $L$-differencing Sperner systems using elementary $p$-adic analysis and polynomial methods, extending classical combinatorial results and addressing open questions.

Contribution

It introduces novel bounds for $q$-modular Sperner systems, extends Snevily's theorem to the $q$-modular setting, and improves previous results on related combinatorial structures.

Findings

Derived new upper bounds on $q$-modular $L$-differencing Sperner systems.

Extended Snevily's theorem to the $q$-modular context.

Provided partial answers to open questions in combinatorics.

Abstract

Let $\mathcal{F}$ be a family of subsets of $[n]$ and $L$ be a subset of $[n]$. We say $\mathcal{F}$ is an $L$-differencing Sperner system if $|A\setminus B|\in L$ for any distinct $A,B\in\mathcal{F}$. Let $p$ be a prime and $q$ be a power of $p$. Frankl first studied $p$-modular $L$-differencing Sperner systems and showed an upper bound of the form $\sum_{i=0}^{|L|}\binom{n}{i}$. In this paper, we obtain new upper bounds on $q$-modular $L$-differencing Sperner systems using elementary $p$-adic analysis and polynomial method, extending and improving existing results substantially. Moreover, our techniques can be used to derive new upper bounds on subsets of the hypercube with restricted Hamming distances. One highlight of the paper is the first analogue of the celebrated Snevily's theorem in the $q$-modular setting, which results in several new upper bounds on $q$-modular $L$-avoiding $L$-intersecting systems. In particular, we improve a result of Felszeghy, Heged\H{u}s, and R\'{o}nyai, and give a partial answer to a question posed by Babai, Frankl, Kutin, and \v{S}tefankovi\v{c}.