  We present an algorithmic framework for computing anti-chains of maximum size
in geometric posets. Specifically, posets in which the entities are geometric
objects, where comparability of two entities is implicitly defined but can be
efficiently tested. Computing the largest anti-chain in a poset can be done in
polynomial time via maximum-matching in a bipartite graph, and this leads to
several efficient algorithms for the following problems, each running in
(roughly) $O(n^{3/2})$ time: (A) Computing the largest Pareto-optimal subset of
a set of $n$ points in $\mathbb{R}^d$. (B) Given a set of disks in the plane,
computing the largest subset of disks such that no disk contains another. This
is quite surprising, as the independent version of this problem is
computationally hard. (C) Given a set of axis-aligned rectangles, computing the
largest subset of non-crossing rectangles.
