Proposition 42 (⇐), p. 28 — a (d+1)-dimensional pseudo-cube B gives a d-dimensional good complex C(B)

Let $B \subseteq \mathcal Y^{d+1}$ be a pseudo-cube of dimension $d + 1$. Then the complex $C(B)$, whose faces are the subsets of the sets $\{(y_i, i) : i \in [d+1]\}$ for $(y_1, \dots, y_{d+1}) \in B$, is a good simplicial complex of dimension $d$.

This is the converse half of the equivalence between pseudo-cubes and good complexes; together with the first half it shows that every concept class of this kind is a properly colored good complex and conversely.

Formalization Note The paper states the converse for a "$d$-dimensional pseudo-cube" and a "$(d-1)$-dimensional" complex; it is indexed here by $d + 1$ and $d$ to avoid natural-number subtraction (at $d = 0$ the page's complex would be "$(-1)$-dimensional"). The vertex type is $\mathcal Y \times$ Fin (d + 1); only the vertices that occur in faces matter. Replacement is the version with a new vertex.