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.
Forbidding and forbidding have the same Erdős–Hajnal constants
Example
The -free class and the -free class have exactly the same Erdős–Hajnal constants. Here is an edge together with an isolated vertex.
Facts & Assumptions
Given: The three-vertex path .
A hereditary class and its complement class have exactly the same Erdős–Hajnal constants (A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants).
The graph has two consecutive edges on three vertices (Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices).
A graph is -free exactly when is -free ( is -free if and only if is -free).
Every fixed-pattern-free graph class is hereditary (Every class defined by forbidden induced subgraphs is hereditary).
Verification
Complementing the two edges of leaves the edge joining its endpoints and makes its middle vertex isolated.
By [L3], complementation maps the -free class bijectively to the -free class; both are hereditary by [L4].
The equality of their constant sets now follows from [L1].
Depends on
- A hereditary class has the Erdős–Hajnal property exactly when its complementary class does, with the same constants
- Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices
- $G$ is $H$-free if and only if $\overline G$ is $\overline H$-free
- Every class defined by forbidden induced subgraphs is hereditary
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
17 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
- Erdos-Hajnal properties in graphs and hypergraphs, introduction (standard reference, not scraped)