Abstract
If T 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; F ϵ T}. P. Erdös (oral communication) transmitted to me in Nice the following question: Is it true that if T 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| = 2 n 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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