LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passjudge 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.

A wide integral geometric layer forces the complete-or-anticomplete property-(*) blockade

Statement

Assume the structural comb-partition hypothesis and let $c \in (0, 1]$ be a common Erdős–Hajnal constant for $F_{1}$ -free and $F_{2}$ -free graphs. In one decreasing structural partition, let $C_{r}$ be an integral geometric layer with $r < q$ . If every block of $C_{r}$ has size at least $w / ℓ^{5 r / 2}$ , then $G$ has a complete or anticomplete $(k, w / k^{10 / c})$ -blockade for some $k \geq ℓ^{c r / 4}$ .

Facts & Assumptions

Given: $r < q$ , a wide layer $C_{r}$ , and a common constant $c \in (0, 1]$ .

[F1]

The first $m_{r}$ structural blocks form an induced subgraph of the $F_{2}$ -free pattern graph (The structural comb-partition hypothesis).

[F2]

The cutoff bound is $m_{r} \geq ℓ^{r / 4}$ (Integral geometric layers exist, cover the partition, and retain the required cutoff bounds).

[F3]

An Erdős–Hajnal constant $c$ supplies a pattern clique or stable set of size at least $m_{r}^{c}$ in a nonempty $F_{2}$ -free graph (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class).

[F4]

A clique or stable set in a pure-blockade pattern lifts to a complete or anticomplete blockade with the same selected width (Homogeneous sets in pure-blockade patterns lift to complete or anticomplete blockades).

[F5]

Positive real powers obey the iterated-power law (The exponent, product, quotient, and iterated-power laws for positive real bases and real exponents).

Proof

technique · direct

1.1

The induced pattern on the first $m_{r}$ blocks is $F_{2}$ -free: an induced forbidden copy there would also be one in the full pattern. By [F1] and [F3], it has a clique or stable set $S$ of cardinality $k \geq m_{r}^{c}$ .

F1F3

2.1

From [F2] and step 1.1, $k \geq (ℓ^{r / 4})^{c} = ℓ^{c r / 4}$ by [F5].

F2F5step 1.1

2.2

The blocks indexed by $S$ lie among the first $m_{r}$ blocks and therefore in layers through $C_{r}$ ; decreasing block sizes and the width assumption on $C_{r}$ give them size at least $w / ℓ^{5 r / 2}$ . By [F4] they form a complete or anticomplete blockade of length $k$ and at least that width.

F4step 1.1

3.1

Step 2.1 and [F5] give $k^{10 / c} \geq ℓ^{5 r / 2}$ , so $w / ℓ^{5 r / 2} \geq w / k^{10 / c}$ . Together with step 2.2 this proves the claim.

step 2.1step 2.2F5algebra∎

Depends on

Used by

The structural comb-partition criterion implies property (*) Theorem

Dependency tree · two levels

30 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, Claim 5.1.2 (standard reference, not scraped)