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# On the density of sets of divisors

Discrete Mathematics 01/1995; 137(s 1–3):345–349. DOI:10.1016/0012-365X(93)E0114-J
Source: DBLP

ABSTRACT 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.

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ABSTRACT: If is a family of sets and A some set we denote by T ∩ A the following family of subsets of A: T ∩ A = {F ∩ A; FϵT}. P. Erdös (oral communication) transmitted to me in Nice the following question: Is it true that if is a family of subsets of some infinite set S then either there exists to each number n a set A ⊂ S with |A| = n such that |T ∩ A| = 2n or there exists some number N such that |T ∩ A| ⩽ |A|c for each A ⊂ S with |A| ⩾ N and some constant c? In this paper we will answer this question in the affirmative by determining the exact upper bound. (Theorem 2).1
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