Antichains in Products of Linear Orders

Abstract

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 ⁿ has no (incomparability-)antichain of cardinality λ, while L ⁿ⁺¹ has an antichain of cardinality λ.

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