Sperner Properties for Groups and Relations

Abstract

An equivalence on the family of subsets of an e-element set E is hereditary if |a| = |b| and | x {⊆ a : x ∼}|=| x {⊆ b : x ∼}| whenever a , b , c , ⊂ E and a ˜ b. Let W i ˜ denote the number of blocks of ˜ consisting of i -element sets. Setting e =⌊1/2 e ⌋ we prove W 0 ~ ≤ ⋯ ≤ W e ~ and W p ~ ≤ W e − p ~ for all p ≼ e ′. The equivalence ˜ is symmetric ( is selfdual ) if W p ~ = W e − p ~ for all p (if a ∼ b ⇔ E \ a ∼ E \ b ). We prove ˜ is symmetric if ˜ is selfdual. The set of blocks of ˜ has a natural order with X ≼ Y if x ⊆ y for some x ∈ X and y ∈ Y . We study the properties of this order, in particular, we prove that for ˜ symmetric the order has the strong Sperner property: for all k the union of the k largest levels is a maximum sized k-family (i.e. a maximum sized union of k antichains). For a permutation group G on E put a ˜ G b if b = g(a) for some g ∈ G . This set-orbit partition is symmetric and therefore the associated order has the strong Sperner property. A direct application proves that the following finite orders have the strong Sperner property: (a) product of chains ( De Bruijn et al., 1949 ) and (b) the initial segments of the product of two chains ( Stanley, 1980 ). Another consequence is that among the unlabelled graphs on n vertices the graphs with ⌊ 1 2 ( n 2 ) Ȱ edges form a maximum sized family allowing no embedding (as a subgraph) between its members. For a binary relation R set a ˜ R b if R ⋂ a 2 (i.e. the restriction of R to the set a ) is isomorphic to R ⋂ b 2 . This equivalence is hereditary. Its equivalence classes are essentially the isomorphism types of restrictions of R and the above order is the usual embedability order of isomorphism types. A consequence of the main result is that for a homogeneous R the order has the strong Sperner property.