CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-16

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Three pair densities equal to $1 / 2$ need not produce a single transversal triangle

Statement

There are three pairwise disjoint vertex sets $X, Y, Z$ for which every cross-density equals $1 / 2$ but no transversal triple spans a triangle.

Facts & Assumptions

Given: Three nonempty even-sized sets, each split equally into parts labelled $0$ and $1$ .

[L1]

Cross-density is the proportion of possible cross-pairs that are edges (Edge counts and densities between nonempty vertex sets).

[L2]

The triangle counting lemma requires regularity in addition to positive pair densities (Triangle counting lemma for three pairwise regular vertex sets).

Counterexample

technique · contradiction

1.1

Join $X$ to $Y$ and $Y$ to $Z$ exactly when the endpoint labels agree, and join $X$ to $Z$ exactly when their labels differ.

givenchoose

2.1

For each cross-pair and each vertex, exactly half the vertices on the other side are neighbours. Thus all three densities are $1 / 2$ by [L1].

step 1.1L1algebra

2.2

Suppose, for contradiction, that $(x, y, z)$ is a transversal triangle. Its $X Y$ and $Y Z$ edges force the three labels to satisfy $x = y = z$ , while its $X Z$ edge forces $x \neq z$ .

assume-contrastep 1.1

3.1

This contradiction shows that no transversal triangle exists. Therefore density alone does not imply the conclusion of [L2]; its regularity hypothesis carries real content.

step 2.2L2discharge-contradiction∎

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: 8 results over 8 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 2.2.1 and Remark 2.2.2 (standard reference, not scraped)

Three pair densities equal to 1/21/2 need not produce a single transversal triangle