I'll present joint work with Emanuele Frittaion about the reverse
mathematics of theorems about partial orders included in Fra?ssé's
book 'Theory of Relations' and originally due to Milner and Pouzet,
Bonnet, and Erdös and Tarski. Some theorems deal with the existence
of linear extensions that preserve some finiteness property, while
the others are concerned with the structure and cardinality of the
collection of initial intervals in partial orders without infinite
antichains. We obtain that some statements are equivalent to ACA_0,
others to ATR_0, and others to Sigma^0_2-bounding. We have also
statements for which we do not know the precise answer yet: they are
provable in WKL_0 but not in RCA_0.