A poset is well‐partially ordered (WPO) if all its linear extensions are well orders; the supremum of ordered types of these linear extensions is the length, of p. We prove that if the vertex set X is infinite, of cardinality κ, and the ordering ⩽ is the intersection of finitely many well partial orderings of X,, then, letting, with, denote the euclidian division by κ (seen as an initial ordinal) of the length of each corresponding poset: where denotes the least (...) initial ordinal greater than the ordinal. This inequality is optimal. This result answers questions of Forster. (shrink)

The age of a relational structure R is the set A of finite restrictions of R considered up to isomorphism. R. Fraı̈ssé, who introduced this notion, showed that ages coincide with nonempty ideals of the poset consisting of finite relational structures, considered up to isomorphism and ordered by embeddability. Here, given two ages A ⊆ B , we study the poset D consisting of ages C in sandwich between A and B . Among other things we show that if D (...) is infinite then it contains an infinite chain. (shrink)