ExampleConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Two complete pairs and one anticomplete pair produce exactly $∣ X_{1} ∣ ∣ X_{2} ∣ ∣ X_{3} ∣$ induced copies of $P_{3}$

Statement

Let $X_{1}, X_{2}, X_{3}$ be disjoint nonempty vertex sets. If $(X_{1}, X_{2})$ and $(X_{2}, X_{3})$ are complete and $(X_{1}, X_{3})$ is anticomplete, then exactly $∣ X_{1} ∣ ∣ X_{2} ∣ ∣ X_{3} ∣$ 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 $P_{3}$ has edges $12, 23$ and nonedge $13$ (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Verification

technique · direct

1.1

Every triple $(x_{1}, x_{2}, x_{3})$ in the product has edges $x_{1} x_{2}, x_{2} x_{3}$ and nonedge $x_{1} x_{3}$ . By [L3] it induces the labelled path $1 - 2 - 3$ .

givenL3

2.1

Conversely every part-respecting labelled triple is one of these product choices, so their number is exactly $∣ X_{1} ∣ ∣ X_{2} ∣ ∣ X_{3} ∣$ .

step 1.1algebra

3.1

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].

givenL1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 22 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Y. Zhao, Graph Theory and Additive Combinatorics, Theorem 2.6.2 with Remark 2.6.3(b) (standard reference, not scraped)

Two complete pairs and one anticomplete pair produce exactly ∣X1∣∣X2∣∣X3∣∣X1​∣∣X2​∣∣X3​∣ induced copies of P3P3​