Shadows and isoperimetry under the sequence-subsequence relation

Some New Results on Macaulay Posets

“…It was shown by Harper [35] that, among all subsets X ⊆ B i of fixed cardinality, the IS in the VIP-order minimizes |Γ B (X)| (the size of the vertex-boundary of X in the Boolean lattice B i ). This order induces a total order of the elements for each level of SO (2). For convenience, we define w(x 1 x 2 .…”

“…Theorem 19 (Ahlswede, Cai [2], Daykin, Danh [24,25], Bezrukov [9]). (SO(2), ≤, vip ) is a Macaulay structure.…”

“…Based on the numerical approach of Ahlswede and Cai in [2], Engel and Leck [31] provided a relatively simple proof of Theorem 19. One of the main observations relates the SMP for SO(2) to the VIP for Boolean lattices: If X ⊆ N i (SO (2)) is a final segment, then |∇(X)| = |Γ B (X)| + 2|X| holds.…”

Shadows under the word-subword relation

“…Quite surprisingly, it seems that the minimal shadow problem for the word-subword relation introduced here has not been studied before, whereas its analogs for sets [1][2][3][4], sequences [5], and vector spaces over finite fields [6] are well known.…”