ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-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.

The $P_{3}$ -free case is much stronger than the general lower bounds

Example

For $P_{3}$ -free graphs the general lower bounds of this page are far from sharp.

Facts & Assumptions

Given: A nonnull finite $P_{3}$ -free graph $G$ with $n : = ∣ V (G) ∣$ .

[L1]

Every $P_{3}$ -free graph satisfies $hom (G) \geq \sqrt{n}$ (Every $P_{3}$ -free graph $G$ satisfies $hom (G) \geq \sqrt{∣ V (G) ∣}$ ).

[L2]

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

Verification

technique · direct

1.1

By [L1], the $P_{3}$ -free class admits the lower bound $hom (G) \geq \sqrt{n} = 2^{(\log_{2} n) / 2}$ .

L1L2algebra

2.1

The exponent $(\log_{2} n) / 2$ grows faster than every constant multiple of $\sqrt{\log_{2} n}$ , because if $L : = \log_{2} n$ and $L \geq 4 a^{2}$ , then $L / 2 \geq a \sqrt{L}$ . Hence for all sufficiently large $n$ , $2^{(\log_{2} n) / 2} \geq 2^{a \sqrt{\log_{2} n}}$ for every fixed $a > 0$ .

step 1.1algebra

2.2

The same exponent $(\log_{2} n) / 2$ also grows faster than every constant multiple of $\sqrt{\log_{2} n \log_{2} \log_{2} n}$ , because for $L \geq 16$ one has $\log_{2} L \leq \sqrt{L}$ , so $\sqrt{L \log_{2} L} \leq L^{3 / 4}$ and then $L / 2 \geq b L^{3 / 4}$ for all sufficiently large $L$ . Thus for every fixed $b > 0$ and all sufficiently large $n$ , $2^{(\log_{2} n) / 2} \geq 2^{b \sqrt{\log_{2} n \log_{2} \log_{2} n}}$ .

step 1.1algebra

3.1

So the square-root homogeneous-set bound for $P_{3}$ -free graphs is much stronger than either general scale on this page.

step 2.1step 2.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

24 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

Maria Chudnovsky, The Erdős-Hajnal Conjecture: A Survey, sec. 2 (standard reference, not scraped)

The P3P3​-free case is much stronger than the general lower bounds