1092.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

3 declarations, 2 contain sorry, 0 declared as axioms

Erdős Problem 1092

References:

Erdős Problem 1092
[Ro82] V. Rödl, Nearly bipartite graphs with large chromatic number, Combinatorica 2 (1982), 377–383.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def Erdos1092.IsAdmissible An edge-budget function line 51 source
Source documentation
An edge-budget function $g$ is admissible for $r$ if every graph $G$ with the property that every subgraph $H$ on $m$ vertices is the union of a graph with chromatic number $\leq r$ and a graph with $\leq g(m)$ edges has chromatic number $\leq r+1$.
The quantification is over all finite graphs $G$ (of any size), not just graphs on a fixed vertex set, and the condition on subgraphs is imposed at every size $m$ simultaneously.
The function $f_r$ of the source is "maximal" among admissible functions. Since a function that is pointwise at most an admissible function is again admissible, $f_r(n) \gg n$ means that some admissible function grows at least linearly.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SimpleGraph, Fin, SimpleGraph.Subgraph, Exists, Finset, Sym2, SimpleGraph.Subgraph.verts, And, LE.le, SimpleGraph.edgeFinset, SimpleGraph.Subgraph.coe, Membership.mem, Set, Classical.propDecidable, SimpleGraph.Subgraph.Adj, Finset.card, Fintype.card, ENat, SimpleGraph.chromaticNumber, SimpleGraph.deleteEdges and 2 more
21 instances, coercions and auxiliary constants left out.
```
def IsAdmissible (r : ℕ) (g : ℕ → ℕ) : Prop :=
…
```
theorem Erdos1092.f_asymptotic_2 Is it true that $f_2(n) \gg n$? sorry line 68 source
Source documentation
Is it true that $f_2(n) \gg n$? Disproved by Rödl, who showed $f_r(n) = o(n)$ for all fixed $r \geq 2$. A conjecture of Erdős, Hajnal, and Szemerédi.
This seems to be closely related to, but distinct from, 744.
Tang notes in the comments that Rödl [Ro82] constructed, for any $\epsilon>0$ and $k$, a graph with chromatic number $\geq k$ such that every graph on $m$ vertices is bipartite after deleting at most $\epsilon m$ edges.

L4DEL explanation
Is it true that $f_2(n) \gg n$? Disproved by Rödl, who showed $f_r(n) = o(n)$ for all fixed $r \geq 2$. A conjecture of Erdős, Hajnal, and Szemerédi. This seems to be closely related to, but distinct from, 744. Tang notes in the comments that Rödl [Ro82] constructed, for any $\epsilon>0$ and $k$, a graph with chromatic number $\geq k$ such that every graph on $m$ vertices is bipartite after deleting at most $\epsilon m$ edges.
Sources and attribution: Pinned Lean source (1646ca16afd6)

depends on sorry what this means
Statement uses: Iff, False, Exists, Nat, And, Erdos1092.IsAdmissible, OfNat.ofNat, Asymptotics.IsBigO, Real, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem f_asymptotic_2 : answer(False) ↔
    ∃ g : ℕ → ℕ, IsAdmissible 2 g ∧
      (fun n : ℕ => (n : ℝ)) =O[atTop] (fun n : ℕ => (g n : ℝ)) := by
…
```
Mentions: Erdos1092.IsAdmissible
theorem Erdos1092.f_asymptotic_general More generally, is $f_r(n)\gg_r n$ for every $r \geq 2$? sorry line 76 source
Source documentation
More generally, is $f_r(n)\gg_r n$ for every $r \geq 2$? Disproved by Rödl, who showed $f_r(n) = o(n)$ for all fixed $r \geq 2$.

depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Exists, And, Erdos1092.IsAdmissible, Asymptotics.IsBigO, Real, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
12 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem f_asymptotic_general : answer(False) ↔
    ∀ r : ℕ, 2 ≤ r → ∃ g : ℕ → ℕ, IsAdmissible r g ∧
      (fun n : ℕ => (n : ℝ)) =O[atTop] (fun n : ℕ => (g n : ℝ)) := by
…
```
Mentions: Erdos1092.IsAdmissible