A polynomial upper bound for poset saturation

Given a finite poset P, we say that a family F of subsets of [n] is P-saturated if F does not contain an induced copy of P, but adding any other set to F creates an induced copy of P. The induced saturation number of P, denoted by sat^∗(n, P), is the size of the smallest P-saturated family with ground set [n]. In this paper we prove that the saturation number for any given poset grows at worst polynomially. More precisely, we show that sat^∗(n, P) = O(n^c), where c ≤ |P|²/4+1 is a constant depending on P only. We obtain this result by bounding the VC-dimension of our family.