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

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 F-0, ..., F-k such that F-0 subset of F-1 subset of ... subset of F-k. 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 subset of X with r - k < vertical bar F vertical bar <= 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 >= r(2) + 4r + 1. Two families .F and g are called cross-intersecting if F boolean AND G not equal empty set for all F is an element of F, G is an element of g. What is the maximum of vertical bar F vertical bar + vertical bar g vertical bar 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. (C) 2020 Elsevier Ltd. All rights reserved.