Simplicial order on the discrete cube ( $≺_{simp}$ )

Order subsets first by increasing cardinality and, within each level, by lexicographic order, with the smallest differing coordinate belonging to the earlier set. This convention makes simplicial initial segments the extremizers in Harper theorem.

Table of contents
- Complementary-size simplicial vertex boundaries can be asymmetric Simplicial order on the discrete cube
- Simplicial vertex boundaries need not increase below half volume Simplicial order on the discrete cube

Complementary-size simplicial vertex boundaries can be asymmetric ( $∣ \partial C (63) ∣ = 36 < 37 = ∣ \partial C (65) ∣ (n = 7)$ )

 0  0

In the seven-dimensional Boolean hypercube,

C (63)

contains every set of size at most three except

{5, 6, 7}

. Its external vertex boundary consists of that missing triple and all 35 four-sets. The complementary-size initial segment

C (65)

contains every set of size at most three and

{1, 2, 3, 4}

. Its external vertex boundary has the other 34 four-sets and three five-sets. Thus complementary family sizes do not impose a symmetric boundary profile.

Simplicial vertex boundaries need not increase below half volume

 0  0

In dimension three, the first three vertices are

\emptyset, {1}, {2}

. Their closed vertex neighbourhood has seven vertices. Adding

{3}

does not change that vertex neighbourhood, so the external vertex boundary decreases from four to three even though both family sizes are at most half the cube.