Alphabeta Math

Regular Pairs and Induced Counting — Examples

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✓ 8 results · all verified · 3 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 5 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

Complete and anticomplete disjoint pairs are 0-regular

Counterexample

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

Example

A 0.01-regular pair restricted to two half-sized subsets is 0.02-regular

Example

The trivial partition has energy d(G)², while the singleton partition records every adjacency

Example

The triangle counting lemma is exact for three complete cross-pairs

Example

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

Counterexample

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

Counterexample

Induced removal must permit adding edges as well as deleting them

Sources