TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01‡ 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 3 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 ‡ Strong Perfect Graph Theorem, ‡ Substituting perfect graphs preserves perfection and ‡ Weak Perfect Graph 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.

Every graph on at most five vertices has the Erdős-Hajnal property

Statement

Every finite graph $H$ with $∣ V (H) ∣ \leq 5$ has the Erdős-Hajnal property.

Facts & Assumptions

Given: A finite graph $H$ with $∣ V (H) ∣ \leq 5$ .

[L1]

Every graph on at most four vertices has the Erdős-Hajnal property (Every graph on at most four vertices has the Erdős-Hajnal property).

[L2]

The bull, $C_{5}$ , $P_{5}$ , and $\overline{P_{5}}$ have the Erdős-Hajnal property (The bull graph has the Erdős-Hajnal property, The five-cycle has the Erdős-Hajnal property, The five-vertex path and its complement have the Erdős-Hajnal property).

[L3]

The prime five-vertex graphs are exactly the bull, $C_{5}$ , $P_{5}$ , and $\overline{P_{5}}$ (The prime five-vertex graphs are exactly the bull, $C_{5}$ , $P_{5}$ , and $\overline{P_{5}}$ ).

[L4]

A finite graph with at least two vertices is prime exactly when it is not a nontrivial substitution (A graph with at least two vertices is prime exactly when it is not obtained by substituting one graph on at least two vertices for a vertex of another graph on at least two vertices).

[L5]

Substitution preserves the Erdős-Hajnal property (Alon–Pach–Solymosi: if $H_{1}$ and $H_{2}$ have the Erdős–Hajnal property, so does the graph obtained from $H_{1}$ by substituting $H_{2}$ for a vertex).

[F1]

If $H ≅ H_{1} [v \to H_{2}]$ and $∣ V (H) ∣ = 5$ with $∣ V (H_{1}) ∣, ∣ V (H_{2}) ∣ \geq 2$ , then $∣ V (H_{1}) ∣ + ∣ V (H_{2}) ∣ - 1 = 5$ , so each factor has at most four vertices.

Proof

technique · cases

1.1

[assume-case small] If $∣ V (H) ∣ \leq 4$ , then [L1] gives the result.

1.2

[assume-case five] Assume $∣ V (H) ∣ = 5$ . We distinguish whether $H$ is prime.

givencases

2.1

[assume-case prime] If $H$ is prime, then [L3] shows that $H$ is isomorphic to one of the four graphs listed in [L2]. Therefore $H$ has the Erdős-Hajnal property.

step 1.2L2L3

2.2

[assume-case nonprime] If $H$ is not prime, then [L4] gives a substitution representation $H ≅ H_{1} [v \to H_{2}]$ with $∣ V (H_{1}) ∣, ∣ V (H_{2}) ∣ \geq 2$ . By [F1] both factors have at most four vertices, so [L1] gives the Erdős-Hajnal property for $H_{1}$ and $H_{2}$ . Applying [L5], the graph $H$ also has the Erdős-Hajnal property.

step 1.2L1L4L5F1

3.1

The cases in steps 1.1, 2.1, and 2.2 exhaust all graphs with at most five vertices. Hence every such graph has the Erdős-Hajnal property.

step 1.1step 2.1step 2.2cases-exhaustive∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

44 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, Section 2 (standard reference, not scraped)
Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Section 1 (standard reference, not scraped)