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  "body": "\n## Statement\n\nConsider the on-line game in which the points of a [[partially ordered set|0QV2JS9RJ3]] $P$ are shown to Assigner one at a time.\nWhen a point appears, Assigner sees how it compares with the points shown before it.\nShe must at once put it into a class, and she can never move it later.\nEach class must end up as an [[antichain|03HF498P1J]].\n\nFix an integer $h \\ge 1$.\nThen Assigner has a strategy for this game that works for every poset $P$ whose [[height|MPR6MH4YSQ]] is at most $h$.\nThis strategy splits $P$ into at most $\\binom{h+1}{2}$ antichains.\n\n## Remarks\n\nThe off-line bound is only $h$ antichains, by [[Mirsky's theorem|5VPXR3RDF1]].\nThe theorem shows that the on-line setting costs a factor of about $(h+1)/2$ in the number of classes.\n",
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