Alphabeta MathHow 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 a single graph, the Erdős–Hajnal property, the polynomial Rödl property, and virality are equivalent
Statement
For every finite graph , the following are equivalent:
- has the Erdős–Hajnal property.
- The singleton family has the polynomial Rödl property.
- The singleton family is viral.
Facts & Assumptions
Given: A finite graph .
The finite-family equivalence theorem applies to every finite family, in particular to the singleton family (For a finite family, the Erdős–Hajnal property, the polynomial Rödl property, and virality are equivalent).
A graph is -free exactly when it is -free (-free and -free graphs under the induced-subgraph convention).
The meanings of “ has the Erdős–Hajnal property”, “the singleton family has the polynomial Rödl property”, and “the singleton family is viral” are those of the corresponding definitions (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class, The polynomial Rödl property for a finite forbidden family, The viral property for a finite forbidden family).
Proof
Applying [L1] to the family gives the equivalence of the finite-family Erdős–Hajnal property, the finite-family polynomial Rödl property, and virality for .
By [L2] and [L3], the first of those assertions is exactly “ has the Erdős–Hajnal property”, while the other two are already assertions about the singleton family .
Steps 1.1 and 1.2 give the claimed equivalence.
Depends on
- For a finite family, the Erdős–Hajnal property, the polynomial Rödl property, and virality are equivalent
- The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class
- The polynomial Rödl property for a finite forbidden family
- The viral property for a finite forbidden family
- $H$-free and $\mathcal F$-free graphs under the induced-subgraph convention
Used by
- The polynomial Rödl witness need not be the whole graph Counterexample
- The singleton family {P₃} is viral Example
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
- M. Bucić, J. Fox, and H. T. Pham, Equivalence between Erdős-Hajnal and polynomial Rödl and Nikiforov conjectures, Theorem 4 (standard reference, not scraped)