Finding one excellent element in case of one lie

TL;DR

This paper investigates the problem of identifying at least one 'excellent' element in an unknown set using subset queries with at most one incorrect answer, aiming to find the minimal such query family.

Contribution

It introduces a method to determine the smallest family of subset queries capable of reliably finding an excellent element despite a single possible error in responses.

Findings

Established the minimal family size for the problem.

Provided algorithms for constructing such families.

Analyzed the error tolerance in element identification.

Abstract

An n-element set contains an unknown number of excellent elements, and our goal is to identify at least one of these elements. The members of a family of subsets can be asked if they contain at least one excellent element or not. At most one of the answers can be wrong. We find the smallest family that finds one excellent element or claims that there is none.