Minimal families in terms of double-sided shadow in the Boolean cube layer

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

“…Since [n] k is isomorphic to [n] n−k , the same result holds when n 2 < k ≤ n−4. It is known [3] that there is a nested system of minimal sets contained in C 1 (n, k) ∪ {{1, 2, . .…”

“…There does not exist a minimizing order on [8] 4 , and the maximal size m of a nested system of minimal families equals 42, since the only minimal ideals having size 41 are I({2, 4, 5, 8}) ∪ I({2, 3, 7, 8}) and I({2, 3, 6, 8}) ∪ I({3, 4, 5, 8}), while the only minimal ideal having size 42 is I ({2, 4, 6, 8}). There does not exist a minimizing order on [7] 3 , and m = 15, since the only minimal ideal having size 15 is I({1, 6, 7}), and it does not contain C 1 (7,3). Finally, there is a minimizing order on [6] 3 distinct from lexicographical: {1, 2, 3} < min {1, 2, 4} < min {1, 2, 5} < min {1, 2, 6} < min {1, 3, 4} < min {1, 3, 5} < min {1, 3, 6} < min {2, 3, 4} < min {2, 3, 5} < min {2, 3, 6} < min {1, 4, 5} < min {1, 4, 6} < min {2, 4, 5} < min {2, 4, 6} < min {1, 5, 6} < min {2, 5, 6} < min {3, 4, 5} < min {3, 4, 6} < min {3, 5, 6} < min {4, 5, 6}.…”