AbstractA 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.