Cardinalities of infinite antichains in products of chains (Q5950768)

M. Ploščica and M. Haviar posed in 1998 the following problem: Is there a chain \(L\) such that \(L^3\) has an infinite antichain of cardinality \(\kappa \) while \(L^2\) does not? In response, the author constructs such a chain \(L\) for any singular \(\kappa \). Independently, such chains have been constructed for certain regular \(\kappa \) recently by M. Goldstern and S. Shelah.