Nonexistence of a Kruskal–Katona type theorem for double-sided shadow minimization in the Boolean cube layer

A double-sided shadow minimization problem in the Boolean cube layer is investigated in this paper. The problem is to minimize the size of the union of the lower and upper shadows of a k-uniform family of subsets of [n]. It is shown that if 3 ≤ k ≤ n− 3, there is no total order such that all its initial segments have minimal double-sided shadow. Denote by ([n] k ) the family of all subsets of the set [n] = {1, 2, . . . , n} having the size k. Let F ⊆ ([n] k ) . The lower shadow ∆F is the (k − 1)-uniform family of sets A such that there exists B ∈ F , A ⊂ B. Similarly, the upper shadow ∇F is the (k + 1)-uniform family of sets A such that there exists B ∈ F , A ⊃ B. The double-sided shadow onF is the union of the families ∆F and ∇F . A family F ⊆ ([n] k ) is minimal in terms of lower shadow (upper shadow, double-sided shadow) if |∆F | ≤ |∆G| (corr., |∇F | ≤ |∇G|, | onF | ≤ | onG|) for each family G ⊆ ([n] k ) such that |G| = |F |. A set A lexicographically precedes a set B (A <lex B) if and only if max((A\ B) ∪ (B \A)) ∈ B. Kruskal [8] and Katona [7] described solutions of the single-sided shadow minimization problem: Kruskal–Katona theorem. The initial lexicographical segments of ([n] k ) are minimal families in terms of lower shadow. A simple modern proof of this theorem is given in [1]. Clements and Lindström [6] generalized this result to the products of chains. Analogues of the Computing Classification System 1998: G.2.1 Mathematics Subject Classification 2010: 05D05, 06A07, 68R15