Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16
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The single-forbidden-graph and finite-nonempty-family formulations of the Erdős–Hajnal conjecture are equivalent

Statement

The following assertions are equivalent:

  1. every finite graph H has the Erdős–Hajnal property;
  2. for every finite nonempty family F of finite graphs, the hereditary class of F-free graphs has the Erdős–Hajnal property.

Facts & Assumptions

Given: The two universally quantified assertions in the Statement.

[L1]

The Erdős–Hajnal property of a graph H is the property of its H-free class (The Erdős–Hajnal property and an Erdős–Hajnal constant for a hereditary graph class).

[L2]

A graph is F-free exactly when it is H-free for every H∈F (H-free and F-free graphs under the induced-subgraph convention).

[L3]

The Erdős–Hajnal property passes to hereditary subclasses (The Erdős–Hajnal property and each of its constants pass to hereditary subclasses), and every family-free class is hereditary (Every class defined by forbidden induced subgraphs is hereditary).

Proof

technique · direct
1.1

Assume assertion 1, let F be finite and nonempty, and choose H∈F. By [L2], every F-free graph is H-free.

givenL2choose
1.2

Conversely, assume assertion 2 and let H be any finite graph. Applying assertion 2 to the finite nonempty family {H} gives the property for the H-free class, which is assertion 1 by [L1] and [L2].

givenL1L2
2.1

The H-free class has the property by assertion 1 and [L1], so its hereditary subclass of F-free graphs has it by [L3]. This proves assertion 2.

step 1.1L1L3
3.1

The two implications prove the equivalence.

step 2.1step 1.2∎

Depends on

Used by

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Sources