Ordinal remainders of ψ-spaces

We consider which ordinals, with the order topology, can be Stone–Čech remainders of which spaces of the form ψ(κ,M), where ω⩽κ is a cardinal number and M⊂ω[κ] is a maximal almost disjoint family of countable subsets of κ (MADF). The cardinality of the continuum, denoted c, and its successor cardinal, c+, play important roles. We show that if κ>c+, then no ψ(κ,M) has any ordinal as a Stone–Čech remainder. If κ⩽c then for every ordinal δ<κ+ there exists Mδ⊂ω[κ], a MADF, such that βψ(κ,Mδ)∖ψ(κ,Mδ) is homeomorphic to δ+1. For κ=c+, βψ(κ,Mδ)∖ψ(κ,Mδ) is homeomorphic to δ+1 if and only if c+⩽δ