Shattering-extremal set systems from Sperner families

We say that a set system F ⊆ 2 shatters a given set S ⊆ [n] if 2 = {F ∩ S : F ∈ F}. The Sauer-Shelah lemma states that in general, a set system F shatters at least |F| sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly |F| sets. A conjecture of Rónyai and the second author and of Litman and Moran states that if a family is shattering-extremal then one can add a set to it and the resulting family is still shattering-extremal. Here we prove this conjecture for a class of set systems defined from Sperner families.