On the density of sets of divisors

Consider the lattice of divisors of n, [1, n]. For any downset (ideal) ℐ in [1, n] we get a forbidden configuration theorem of the type that if a set of divisors D avoids certain configurations, then |D|⩾|ℐ|. If we let ℓ be the set of minimal elements of [1, n] not in ℐ, then we forbid in D the configurations C(s) (defined in the paper) for s∈ℒ. This generalizes a result of Alon and in turn generalizes a result of Sauer, Perles and Shelah.