Antichains in Products of Linear Orders

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

• Type: Journal
• Year: 2002
• Where: ORDER
• Authors: Martin Goldstern, Saharon Shelah
• Keywords: Linear Order, ORDER 2002, Regular Cardinals, Singular Strong Limit