Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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-regular vertex partitions, equitable partitions, and refinement
Definition
Let be a nonempty graph and let be a partition of into nonempty parts. For , the partition is -regular if where ordered pairs and the diagonal pairs are included (-regular pairs and self-regular vertex sets).
The partition is equitable if for all . A partition refines if every part of is contained in a part of . The empty family is the unique partition of the null graph and is equitable and -regular by convention.
Depends on
Used by
- A prescribed finite vertex partition has a bounded ε-regular refinement, equitable when the initial partition is equitable Corollary
- The mean-square density, or energy, of a vertex partition Definition
- Every nonregular k-part partition has a refinement with energy gain greater than ε⁵ and at most k2ᵏ⁺¹ parts Lemma
- Equitable strong regularity lemma: a very regular refinement that changes energy only slightly Theorem
- Szemerédi regularity lemma with an equitable partition and an explicit tower-type upper bound for graphs of order at least m₀ Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 4 results over 4 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, Definition 2.1.7 and Theorem 2.1.20 (standard reference, not scraped)
- D. Conlon and J. Fox, Graph removal lemmas, sec. 2.1 (standard reference, not scraped)