CounterexampleConstruction: AI-generatedVerification: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Omitting cross-block purity breaks the transversal conclusion

Statement refuted

It is false that one wide pure-partition block selected from each comb block must form a pure transversal if cross-block purity is omitted.

Facts & Assumptions

Given: Four comb blocks $B_{i} = {b_{i}^{0}, \dots, b_{i}^{7}}$ and teeth $a_{i}$ . Each $a_{i}$ is complete to $B_{i}$ and anticomplete to the other $B_{j}$ . Take the one-block partitions $A_{1}^{i} = B_{i}$ . Put precisely one edge, $b_{1}^{0} b_{2}^{0}$ , between $B_{1}$ and $B_{2}$ , and put no edges between any other distinct pair of blocks.

[F1]

The specified tooth/block incidences make these four pairs an $(4, 8)$ -comb (Combs in a graph).

[F2]

A pair is mixed when it is neither complete nor anticomplete (Edges between disjoint vertex sets; complete, anticomplete, pure and mixed pairs).

Counterexample

technique · direct finite adjacency check

1.1

Each one-block partition $(A_{1}^{i})$ is vacuously a pure blockade, and $∣ A_{1}^{i} ∣ = 8 \geq 8 / (2 \cdot 4) = 1$ ; [F1] confirms that the ambient blocks are a comb.

F1givenalgebra

1.2

The pair $(A_{1}^{1}, A_{1}^{2})$ has the edge $b_{1}^{0} b_{2}^{0}$ but, for example, does not have $b_{1}^{0} b_{2}^{1}$ . It is neither anticomplete nor complete, hence is mixed by [F2].

F2given

2.1

Therefore the four wide selected blocks are not a pure transversal. The missing cross-block-purity condition is exactly what fails here.

step 1.2F2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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

Huang, Ju, and Zhou, Erdős–Hajnal beyond the five-vertex path, condition (2.3) and Claim 5.1.1 (standard reference, not scraped)