Builder strategy forcing many antichains in on-line poset partitioning

Statement

Two players build a partially ordered set $P$ in an on-line game. The builder presents the points of $P$ one at a time, and states how each new point compares with the points shown before. The assigner must put each new point at once into an antichain, and cannot change this later. The antichains together partition $P$.

Then for every integer $h \ge 1$, the builder has a strategy $S_h$ that produces a poset $P$ of height $h$ and forces the assigner to do both of the following.

1. The assigner uses at least $\binom{h+1}{2}$ antichains to partition $P$.
2. The assigner uses at least $h$ different antichains for the set of maximal elements of $P$.