On Fraïssé’s conjecture for linear orders of finite Hausdorff rank

We prove that the maximal order type of the wqo of linear orders of finite Hausdorff rank under embeddability is φ 2 ( 0 ) , the first fixed point of the ε -function. We then show that Fraïssé’s conjecture restricted to linear orders of finite Hausdorff rank is provable in ACA 0 +  +“ φ 2 ( 0 ) is well-ordered” and, over RCA 0 , implies ACA 0 ′  +“ φ 2 ( 0 ) is well-ordered”.