Analogues of Milner’s Theorem for families without long chains and of vector spaces

Abstract

Let n>k>0 be integers and X an n-element set. A family F consisting of subsets of X is called k-Sperner if it has no distinct members F0,…,Fk such that F0⊂F1⊂…⊂Fk. A family is called s-union if the union of any two of its members has size at most s. A classical result of Milner determines the maximum size of a family that is both 1-Sperner and s-union. The present paper is dealing with the case k≥2. If s=2r<n then the natural construction is to take all subsets F⊂X with r−k<|F|≤r. Theorem 4.1 shows that this is optimal for n>r(r+3). The case of s=2r+1 is more complex. We believe that Example 1.9 provides the maximum. Theorem 1.12 confirms this for k=2 and n≥r2+4r+1.Two families F and G are called cross-intersecting if F∩G≠0̸ for all F∈F, G∈G. What is the maximum of |F|+|G| if in addition F is k-Sperner, G is ℓ-Sperner? The exact answer is given by Theorem 1.4.In Section 3 we prove the analogue of Milner’s Theorem for vector spaces.