Searching for quicksand ideals in partially ordered sets

TL;DR

This paper investigates optimal search strategies for identifying unknown ideals in partially ordered sets with limited positive responses, providing tight bounds and specific strategies for certain cases.

Contribution

It introduces tight bounds for the minimal number of queries needed and constructs optimal strategies for the case where two positive responses are allowed in product posets.

Findings

Established tight bounds for $m_k()$

Developed optimal search strategies for $k=2$ in specific product posets

Analyzed the complexity of searching within partially ordered sets

Abstract

We consider a combinatorial question about searching for an unknown ideal $\mu$ within a known poset $\lambda$. Elements of $\lambda$ may be queried for membership in $\mu$, but at most $k$ positive query results are permitted. The goal is to find a search strategy which guarantees a solution in a minimal total number $m_k(\lambda)$ of queries. We provide tight bounds for $m_k(\lambda)$, and construct optimal search strategies for the case where $k=2$ and $\lambda$ is the product poset of totally ordered finite sets, one of which has cardinality not more than six.