DefinitionDefinition: AI-adaptedProof: Not applicableaudited 2026-07-31

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The lower and upper shadows of a uniform set family

Definition

Let $A$ be a finite set and let $F \subseteq [A]^{k}$ be a $k$ -uniform family (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ). Its lower shadow is

$\partial F : = {R \in [A]^{k - 1} : R \subset S for some S \in F}$

when $k \geq 1$ , and $\partial F : = \emptyset$ when $k = 0$ . Its upper shadow is

$\nabla F : = {T \in [A]^{k + 1} : S \subset T for some S \in F} .$

If $k = ∣ A ∣$ , the upper shadow is empty. Both shadows consist of the immediate neighbours of $F$ one rank below or above it in the Boolean lattice.

Depends on

The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$

Used by

Dependency tree · two levels

10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. Keller and W. T. Trotter, Applied Combinatorics, §6.2 (standard reference, not scraped)