An efficient search strategy for hidden ideals in pointed partially ordered sets

We consider a combinatorial question about searching for an unknown ideal $\mu$ within a known pointed poset $\lambda$. Elements of $\lambda$ may be queried for membership in $\mu$, but at most $k$ positive queries are permitted. We provide a general search strategy for this problem, and establish new bounds (based on $k$ and the degree and height of $\lambda$) for the total number of queries required to identify $\mu$. We show that this strategy performs asymptotically optimally on the family of complete $\ell$-ary trees as the height grows.