Abstract
It is proved that if (P⩽) is a poset with no infinite chain and k is a positive integer, then there exist a partition of P into disjoint chains C i and disjoint antichains A 1, A 2, ..., A k, such that each chain C i meets min (k, |C i|) antichains A j. We make a ‘dual’ conjecture, for which the case k=1 is: if (P⩽) is a poset with no infinite antichain, then there exist a partition of P into antichains A i and a chain C meeting all A i. This conjecture is proved when the maximal size of an antichain in P is 2.
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