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

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The half graph has no regularity across its natural bipartition at a fixed small parameter

Statement

Let $X_{n} = {a_{1}, \dots, a_{n}}$ and $Y_{n} = {b_{1}, \dots, b_{n}}$ , with $a_{i} b_{j}$ an edge exactly when $i \leq j$ . For every $n \geq 4$ , the natural pair $(X_{n}, Y_{n})$ is not $1 / 5$ -regular.

Facts & Assumptions

Given: The displayed bipartite half graph.

[L1]

Failure of $ϵ$ -regularity is witnessed by subsets of relative size at least $ϵ$ whose density differs from the full-pair density by more than $ϵ$ ( $ϵ$ -regular pairs and self-regular vertex sets).

[L2]

A bipartite graph has no edges within either of its two specified sides (Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

Counterexample

technique · direct

1.1

The number of cross-edges is $n + (n - 1) + \dots + 1 = n (n + 1) / 2$ , so $d (X_{n}, Y_{n}) = (n + 1) / (2 n)$ .

givenalgebra

1.2

Put $q = ⌈ n / 4 ⌉$ , $A = {a_{1}, \dots, a_{q}}$ , and $B = {b_{n - q + 1}, \dots, b_{n}}$ . Then $∣ A ∣, ∣ B ∣ \geq n / 4 > n / 5$ , and every $a_{i} \in A$ satisfies $i \leq j$ for every $b_{j} \in B$ , so $d (A, B) = 1$ .

givenL2choosealgebra

2.1

For $n \geq 4$ , one has $∣ d (A, B) - d (X_{n}, Y_{n}) ∣ = (n - 1) / (2 n) \geq 3 / 8 > 1 / 5$ . Together with the size bounds in step 1.2, [L1] shows that $(X_{n}, Y_{n})$ is not $1 / 5$ -regular.

step 1.1step 1.2L1algebra∎

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: 10 results over 6 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, sec. 2.1 (standard reference, not scraped)