Title: A theorem from Rival and Sands and reverse mathematics
by Marta Fiori Carones (LMU Munich) as part of Computability theory and applications
In 1980 Ivan Rival and Bill Sands proved that for each infinite poset P with finite width (i.e. such that there is a fixed finite bound on the size of antichains in P) there is an infinite chain C ⊆ P such that each element of P is comparable to none or to infinitely many elements of C. Moreover, if P is countable, C can be found such that each element of P is comparable to none or to cofinitely many elements of C.
We prove that some versions of the previous theorem are equivalent to the Ascending/descending sequence principle or to related known principles of the reverse mathematics zoo.
(Joint work with Alberto Marcone, Paul Shafer and Giovanni Soldà)