Dev/MonadicDependence/NowhereDenseBridge/Contract.lean
| 1 | import Catalog.Sparsity.Preliminaries.Contract |
| 2 | import Catalog.Sparsity.NowhereDense.Contract |
| 3 | import Dev.MonadicDependence.NowhereDense.Contract |
| 4 | |
| 5 | namespace Dev.MonadicDependence.NowhereDenseBridge |
| 6 | |
| 7 | open Catalog.Sparsity.Preliminaries |
| 8 | |
| 9 | /-- Folklore: Mählmann's local nowhere-denseness |
| 10 | (`Dev.MonadicDependence.NowhereDense.IsNowhereDense`, a per-radius |
| 11 | subdivided-clique subgraph bound) is equivalent to the catalog's |
| 12 | shallow-topological-minor nowhere-denseness |
| 13 | (`Catalog.Sparsity.NowhereDense.IsNowhereDense`, bounded `ω_d` at every |
| 14 | depth `d`). |
| 15 | |
| 16 | The two parameter functions are only *functionally* related. The |
| 17 | forward direction (shallow-minor bound ⇒ subdivision bound) is linear |
| 18 | in the parameter: `N_r ≤ ω_{⌈r/2⌉}(C) + 1`. The backward direction |
| 19 | (subdivision bound ⇒ shallow-minor bound) goes through a Ramsey |
| 20 | argument on BFS trees rooted at branch-set centres and yields only a |
| 21 | bound of Ramsey type in the depth parameter. |
| 22 | |
| 23 | The contract is qualitative: the equivalence as propositions on graph |
| 24 | classes, with no quantitative clause on the parameter relation. See |
| 25 | `review.tex` for the rationale and `proof.tex` for the proof sketch. -/ |
| 26 | axiom nowhereDenseBridge (C : GraphClass) : |
| 27 | Dev.MonadicDependence.NowhereDense.IsNowhereDense C ↔ |
| 28 | Catalog.Sparsity.NowhereDense.IsNowhereDense C |
| 29 | |
| 30 | end Dev.MonadicDependence.NowhereDenseBridge |
| 31 |