Stability through non-shadows

TL;DR

This paper investigates families of sets with restricted intersections, proving a conjecture for large n, and introduces a novel linear algebra approach using non-shadows, with implications for stability in combinatorial inequalities.

Contribution

It proves Snevily's conjecture in a stronger form for large n and develops a new linear algebra technique involving non-shadows for stability results.

Findings

Proved Snevily's conjecture for large n

Established stability results for Kleitman's inequality

Introduced a new linear algebra method using non-shadows

Abstract

We study families $\mathcal{F}\subseteq 2^{[n]}$ with restricted intersections and prove a conjecture of Snevily in a stronger form for large $n$. We also obtain stability results for Kleitman's isodiametric inequality and families with bounded set-wise differences. Our proofs introduce a new twist to the classical linear algebra method, harnessing the non-shadows of $\mathcal{F}$, which may be of independent interest.