Forbidden induced partial orders Original Research Article

For each finite partial order P, we consider the size of the set Forbindn (P) of partial orders on n labelled points containing no induced copy of P. We show that |Forbindn(P)| = 2o(n2) unless P has height at least 3, in which case |Forbindn(P)| = 2n2/4+o(n2). We show that |Forbindn(P)|n!cn for some constant c if and only if P is either an antichain or one of ten small partial orders. Between these extremes, we consider the question of which P have |Forbindn(P)| = nO(n).