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  "body": "\n## Statement\n\nTwo players build a [[partially ordered set|0QV2JS9RJ3]] $P$ in an on-line game.\nThe builder presents the points of $P$ one at a time, and states how each new point compares with the points shown before.\nThe assigner must put each new point at once into an [[antichain|03HF498P1J]], and cannot change this later.\nThe antichains together partition $P$.\n\nThen for every integer $h \\ge 1$, the builder has a strategy $S_h$ that produces a poset $P$ of [[height|MPR6MH4YSQ]] $h$ and forces the assigner to do both of the following.\n\n1. The assigner uses at least $\\binom{h+1}{2}$ antichains to partition $P$.\n2. The assigner uses at least $h$ different antichains for the set of [[maximal elements|072G59HW59]] of $P$.\n\n## Remarks\n\nOff-line, a poset of height $h$ needs only $h$ antichains, by [[Mirsky's theorem|5VPXR3RDF1]].\nThe result shows that an on-line assigner can be forced to use about $h^2/2$ antichains.\n",
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