883.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 883

References:

erdosproblems.com/883
[ErSa97] Erdős, P. and Sárközy, G. N., On cycles in the coprime graph of integers. Electron. J. Combin. (1997), Research Paper 8.
[Sa99] Sárközy, G. N., Complete tripartite subgraphs in the coprime graph of integers. Discrete Math. (1999), 227-238.

Imports (1)

FormalConjecturesUtil

def Erdos883.coprimeGraph The coprime graph on $\mathbb{N}$: two integers are joined by an edge if they are coprime line 40 source
Source documentation
The coprime graph on $\mathbb{N}$: two integers are joined by an edge if they are coprime.

standard axioms: Quot.sound, propext what this means
Statement uses: SimpleGraph, Nat
Proof uses: SimpleGraph.fromRel, Nat.Coprime
```
def coprimeGraph : SimpleGraph ℕ :=
…
```
theorem Erdos883.erdos_883.parts.i For $A\subseteq \{1,\ldots,n\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime sorry line 54 source
Source documentation
For $A\subseteq \{1,\ldots,n\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime.
Is it true that if $$|A| > \lfloor n/2 \rfloor + \lfloor n/3 \rfloor - \lfloor n/6 \rfloor$$ then $G(A)$ contains all odd cycles of length $\leq n/3 + 1$?
A problem of Erdős and Sárközy [ErSa97].

depends on sorry what this means
Statement uses: Iff, True, Nat, Finset, LE.le, Finset.Icc, OfNat.ofNat, LT.lt, HSub.hSub, HAdd.hAdd, HDiv.hDiv, Finset.card, Odd, Membership.mem, Set, SimpleGraph.oddCycleLengths, SimpleGraph.induce, Erdos883.coprimeGraph
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_883.parts.i : answer(sorry) ↔
    ∀ (n : ℕ) (A : Finset ℕ),
      A ⊆ Finset.Icc 1 n →
      n / 2 + n / 3 - n / 6 < A.card →
      ∀ l : ℕ, Odd l → 3 ≤ l → l ≤ n / 3 + 1 →
        l ∈ (coprimeGraph.induce (A : Set ℕ)).oddCycleLengths := by
…
```
theorem Erdos883.erdos_883.parts.ii Documented declaration: ii sorry line 72 source
Source documentation
Is it true that, for every $\ell\geq 1$, if $n$ is sufficiently large and $$|A| > \lfloor n/2\rfloor + \lfloor n/3\rfloor - \lfloor n/6\rfloor$$ then $G(A)$ must contain a complete $(1,\ell,\ell)$ tripartite graph on $2\ell+1$ vertices?
The second question was solved by Sárközy [Sa99], who proved this with $\ell \gg \log n/\log\log n$.

depends on sorry what this means
Statement uses: Iff, True, Nat, LE.le, OfNat.ofNat, Filter.Eventually, Finset, Finset.Icc, LT.lt, HSub.hSub, HAdd.hAdd, HDiv.hDiv, Finset.card, SimpleGraph.IsContained, Sigma, Fin, Matrix.vecCons, Nat.succ, Matrix.vecEmpty, SimpleGraph.completeMultipartiteGraph and 3 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_883.parts.ii : answer(True) ↔
    ∀ l : ℕ, 1 ≤ l → ∀ᶠ n : ℕ in atTop, ∀ A : Finset ℕ,
      A ⊆ Finset.Icc 1 n →
      n / 2 + n / 3 - n / 6 < A.card →
      (SimpleGraph.completeMultipartiteGraph (fun i : Fin 3 ↦ Fin (![1, l, l] i))).IsContained
        (coprimeGraph.induce (A : Set ℕ)) := by
…
```
Mentions: Erdos883.erdos_883.parts.i