Antichains in Products of Linear Orders

We show that: (1) For many regular cardinals λ (in particular, for all successors of singular strong limit cardinals, and for all successors of singular ω-limits), for all n ∈ {2, 3, 4, . . .}: There is a linear order L such that L has no (incomparability-)antichain of cardinality λ, while L has an antichain of cardinality λ. (2) For any nondecreasing sequence 〈λn : n ∈ {2, 3, 4, . . .}〉 of infinite cardinals it is consistent that there is a linear order L such that, for all n: L has an antichain of cardinality λn, but no antichain of cardinality λn .