CorollaryStatement: AI-generatedProof: AI-generatedprecheck passaudited 2026-08-26

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.

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

1.1L1algebra

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

2.1step 1.1choose

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 $n$ .

3.1step 1.1step 2.1algebra

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.

4.1step 3.1∎

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 $2^{c \sqrt{\log_{2} n \log_{2} \log_{2} n}}$ .

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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​