It is known [6] that for every function f in the generalized Schur class $$\mathcal{S}_{\kappa } $$ and every nonempty open subset Ω of the unit disk $$\mathbb{D}$$ , there exist points z1,...,z n ∈Ω such that the n × nPick matrix $${\left[ {\frac{{1 - f(z_{i} )f(z_{j} )^{*} }}{{1 - z_{i} \overline{z} _{j} }}} \right]}^{n}_{{j,i = 1}} $$ has κ negative eigenvalues. In this paper we discuss existence of an integer n0 such that any Pick matrix based on z1,...,z n ∈Ω with n ≥ n0 has κ negative eigenvalues. Definitely, the answer depends on Ω. We prove that if $$\Omega = \mathbb{D}$$ , then such a number n0 does not exist unless f is a ratio of two finite Blaschke products; in the latter case the minimal value of n0 can be found. We show also that if the closure of Ω is contained in $$\mathbb{D}$$ then such a number n0 exists for every function f in $$\mathcal{S}_{\kappa }$$ .