Singular Cardinal Combinatorics and Inner Model Theory: March 5-9, 2007: Gainesville, Florida
Assaf Rinot
Antichains in partially ordered sets of singular cofinality
We say that a singular cardinal λ is a prevalent singular cardinal if and only if there exists a family F of size λ with sup{ | A| : A in F } < λ such that any subset of λ of size less than cf(λ) is covered by some element of F.
In their paper from 1981, Milner and Sauer conjectured that any poset P of singular cofinality must contain an antichain of size cf(cf(P)).
We prove their conjecture restricted to the class of all prevalent singular cardinals.
It is an open problem whether there consistently exists a singular cardinal which is not a prevalent singular cardinal.