Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-16
How statement and proof provenance work

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  • 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.
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Two complete pairs and one anticomplete pair produce exactly ∣X1∣∣X2∣∣X3∣ induced copies of P3

Statement

Let X1,X2,X3 be disjoint nonempty vertex sets. If (X1,X2) and (X2,X3) are complete and (X1,X3) is anticomplete, then exactly ∣X1∣∣X2∣∣X3∣ part-respecting labelled triples induce the path 1−2−3. All three cross-pairs are 0-regular.

Facts & Assumptions

Given: Three pure cross-pairs as in the Statement.

[L1]

The induced counting lemma counts maps satisfying every prescribed edge and nonedge relation across regular pairs (Induced counting lemma: regular edge and nonedge pairs force many induced copies).

[L2]

Complete and anticomplete pairs have density 1 and 0, respectively (Edge counts and densities between nonempty vertex sets), and constant-density pure pairs are 0-regular (ϵ-regular pairs and self-regular vertex sets).

[L3]

The graph P3 has edges 12,23 and nonedge 13 (Empty and complete graphs, complete bipartite graphs, and the convention that Pn and Cn have n vertices).

Verification

technique · direct
1.1givenL3

Every triple (x1,x2,x3) in the product has edges x1x2,x2x3 and nonedge x1x3. By [L3] it induces the labelled path 1−2−3.

2.1step 1.1algebra

Conversely every part-respecting labelled triple is one of these product choices, so their number is exactly ∣X1∣∣X2∣∣X3∣.

3.1givenL1L2∎

The two complete pairs have density 1 and the anticomplete pair density 0; every nonempty subpair retains its density. Hence [L2] gives 0-regularity, making this the zero-error model of [L1].

Depends on

Used by

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Dependency tree · two levels

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Sources