CorollaryStatement: AI-generatedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26‡ rests on unproved material (inherited)

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 2 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Fox–Sudakov: a quantitative density form of Rödl's theorem and ‡ Bucić–Nguyen–Scott–Seymour: a log-log quantitative density theorem. Each is recorded with a citation to the literature and 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 fixed $H$ , the log-log scale eventually exceeds every classical scale $2^{c \sqrt{\log_{2} n}}$

Statement

Let $a, b > 0$ . Then there exists $N \geq 2$ such that for every integer $n \geq N$ , $2^{b \sqrt{\log_{2} n \log_{2} \log_{2} n}} \geq 2^{a \sqrt{\log_{2} n}} .$ In particular, for each fixed finite graph $H$ , the log-log lower bound of Every $H$ -free graph has a homogeneous set of size at least $2^{c \sqrt{\log_{2} n \log_{2} \log_{2} n}}$ eventually exceeds every classical scale $2^{a \sqrt{\log_{2} n}}$ .

Facts & Assumptions

Given: Positive reals $a, b$ .

[L1]

For $n > 1$ , $\log_{2} n$ is defined (The logarithm to a positive base other than one).

Proof

technique · direct

1.1

For every integer $n > 2$ , one has $\sqrt{\log_{2} n \log_{2} \log_{2} n} = \sqrt{\log_{2} \log_{2} n} \sqrt{\log_{2} n}$ .

L1algebra

2.1

Choose $N \geq 4$ so that $b \sqrt{\log_{2} \log_{2} n} \geq a$ whenever $n \geq N$ . This is possible because $\log_{2} \log_{2} n$ tends to $+ \infty$ with $n$ .

step 1.1choose

3.1

For every $n \geq N$ , step 2.1 gives $b \sqrt{\log_{2} n \log_{2} \log_{2} n} \geq a \sqrt{\log_{2} n}$ , and exponentiating base $2$ preserves the inequality.

step 1.1step 2.1algebra

4.1

This proves the displayed eventual inequality, and the final sentence is its application with the constant supplied by Every $H$ -free graph has a homogeneous set of size at least $2^{c \sqrt{\log_{2} n \log_{2} \log_{2} n}}$ .

step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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 (standard reference, not scraped)

For fixed HH, the log-log scale eventually exceeds every classical scale 2clog⁡2n2clog2​n​