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 ε {2, 3, 4 ...}: There is a linear order L such that Lⁿ has no (incomparability-)antichain of cardinality λ, while Lⁿ⁺¹ has an antichain of cardinality λ. (2) For any nondecreasing sequence 〈λ_n: n ε {2, 3, 4, ...}〉 of infinite cardinals it is consistent that there is a linear order L such that, for all n: Lⁿ has an antichain of cardinality λ_n, but no antichain of cardinality λ _n⁺.