DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 2026-08-28

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.

A tau-critical graph

Definition

Let $τ > 0$ . A finite graph $G$ is $τ$ -critical when the two conditions below hold.

$κ (G) < ∣ V (G) ∣^{τ}$ .
Every proper induced subgraph $H$ of $G$ satisfies $κ (H) \geq ∣ V (H) ∣^{τ}$ .

Here $κ$ is that of The parameter kappa(G)=alpha(G)omega(G), induced subgraphs are those of Subgraphs, induced subgraphs and spanning subgraphs, and the real power $∣ V (G) ∣^{τ}$ is that of Real powers for positive bases, with the zero-base positive-exponent convention. The first clause forces every $τ$ -critical graph to be nonempty, because it would read $0 < 0^{τ} = 0$ on the null graph.

Depends on

Used by

Dependency tree · two levels

15 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, Alex Scott, Paul Seymour, and Sophie Spirkl, Erdos-Hajnal for graphs with no 5-hole, before Theorem 3.1 (standard reference, not scraped)