ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-26‡ rests on unproved material

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.

‡ Rests on 1 statement not proved in this library. Every dependency marked ‡ below is recorded with a citation but is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

For large $n$ , the log-log choice of $x$ still leaves a dense-or-sparse set of order at least $\sqrt{n}$

Example

Fix a finite graph $H$ , let $C_{H} > 0$ be the constant from Bucić–Nguyen–Scott–Seymour: a log-log quantitative density theorem ‡, and let $G$ be an $H$ -free graph on $n$ vertices. Write $L : = \log_{2} n$ , and set $β : = 1 / (4 \sqrt{C_{H}})$ and $x : = 2^{- β \sqrt{L \log_{2} L}}$ .

Facts & Assumptions

Given: The data in the Example.

‡ [L1]

For $0 < x < 1 / 2$ , every $H$ -free $n$ -vertex graph has a vertex set of order at least $2^{- C_{H} (\log_{2} (1 / x))^{2} / \log_{2} \log_{2} (1 / x)} n$ whose induced graph or complement has at most $x (\binom{∣ S ∣}{2})$ edges (Bucić–Nguyen–Scott–Seymour: a log-log quantitative density theorem ‡).

Verification

technique · direct

1.1

For the chosen $x$ , one has $\log_{2} (1 / x) = β \sqrt{L \log_{2} L}$ .

givenalgebra

2.1

For all sufficiently large $L$ , the inequality $β \sqrt{L \log_{2} L} \geq \sqrt{L} > 1$ holds. Hence $0 < x < 1 / 2$ and $\log_{2} \log_{2} (1 / x) \geq \frac{1}{2} \log_{2} L$ .

step 1.1algebra

3.1

Hence, for all sufficiently large $n$ , $C_{H} (\log_{2} (1 / x))^{2} / \log_{2} \log_{2} (1 / x) \leq 2 C_{H} β^{2} L = L / 8$ , and therefore $2^{- C_{H} (\log_{2} (1 / x))^{2} / \log_{2} \log_{2} (1 / x)} n \geq 2^{- L / 8} n = 2^{7 L / 8} \geq \sqrt{n}$ .

step 2.1‡ L1algebra

4.1

So for all sufficiently large $n$ , this choice of $x$ still leaves a dense-or-sparse set of order at least $\sqrt{n}$ .

step 3.1‡ L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Matija Bucić, Tung Nguyen, Alex Scott, and Paul Seymour, Induced subgraph density. I. A loglog step towards Erdős-Hajnal, Theorem 1.8 (standard reference, not scraped)

For large nn, the log-log choice of xx still leaves a dense-or-sparse set of order at least nn​