New bounds on Simonyi's conjecture

TL;DR

This paper improves the upper bound on the size product of recovering pairs, a special set system pair, from 2.3264^n to 2.284^n, advancing towards proving Simonyi's conjecture.

Contribution

The paper presents a new combinatorial proof that tightens the upper bound on the size product of recovering pairs, moving closer to the conjectured limit.

Findings

Improved upper bound to 2.284^n for the size product of recovering pairs.

The proof is purely combinatorial, avoiding probabilistic methods.

Progress towards confirming Simonyi's conjecture.

Abstract

We say that a pair $(\mathcal{A},\mathcal{B})$ is a recovering pair if $\mathcal{A}$ and $\mathcal{B}$ are set systems on an $n$ element ground set, such that for every $A,A' \in \mathcal{A}$ and $B,B' \in \mathcal{B}$ we have that ($A \setminus B = A' \setminus B'$ implies $A=A'$) and symmetrically ($B \setminus A = B' \setminus A'$ implies $B=B'$). G. Simonyi conjectured that if $(\mathcal{A},\mathcal{B})$ is a recovering pair, then $|\mathcal{A}||\mathcal{B}|\leq 2^n$. For the quantity $|\mathcal{A}||\mathcal{B}|$ the best known upper bound is $2.3264^n$ due to K\"orner and Holzman. In this paper we improve this upper bound to $2.284^n$. Our proof is combinatorial.