DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-06

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.

Integral geometric layers of a decreasing block partition

Definition

Let $(A_{1}, \dots, A_{t})$ be a partition into nonempty blocks with $∣ A_{1} ∣ \geq \dots \geq ∣ A_{t} ∣$ , where $t \geq ℓ \geq 4$ . For each integer $r \geq 1$ , put $m_{r} : = \max {n \in N : 1 \leq n \leq t and n \leq ℓ^{r / 2}} .$ The set is nonempty because $1 \leq ℓ^{r / 2}$ , and finite, so this maximum is an integer. Let $q$ be the least $r \geq 1$ for which $m_{r} = t$ . The integral geometric layers are $C_{1} : = (A_{1}, \dots, A_{m_{1}}), C_{r} : = (A_{m_{r - 1} + 1}, \dots, A_{m_{r}}) (2 \leq r \leq q) .$

Thus every index used here is integral; the layers are consecutive portions of the original ordered partition. Real powers are those in The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents, and a block has the nonempty meaning of Blockades, their length, their width, and their support.

Depends on

Used by

Dependency tree · two levels

6 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

Huang, Ju, and Zhou, Erdős–Hajnal beyond the five-vertex path, proof of Lemma 5.1 (standard reference, not scraped)