195.lean`

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

3 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 195

References:

erdosproblems.com/195
[Ad22] Adenwalla, S., Avoiding Monotone Arithmetic Progressions in Permutations of Integers. arXiv:2211.04451 (2022).
[Ge19] Geneson, Jesse, Forbidden arithmetic progressions in permutations of subsets of the integers. Discrete Math. (2019), 1489-1491.

Imports (1)

FormalConjecturesUtil

theorem Erdos195.erdos_195 What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1 < \cdots < x_k$? sorry line 45 source
Source documentation
What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1 < \cdots < x_k$?
Here a permutation of $\mathbb{Z}$ is a one-sided arrangement $a_1, a_2, a_3, \ldots$ of the integers, i.e. a bijection $\mathbb{N} \to \mathbb{Z}$, and a monotone $k$-term arithmetic progression is a subsequence $a_{i_1}, \ldots, a_{i_k}$ with $i_1 < \cdots < i_k$ forming an increasing or decreasing arithmetic progression.

depends on sorry what this means
Statement uses: Eq, Nat, sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, SupSet.sSup, Set.ofPred, Equiv, Int, HasMonotoneAP
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_195 :
    answer(sorry) = sSup {k : ℕ | ∀ f : ℕ ≃ ℤ, HasMonotoneAP f k} := by
…
```
Mentions: HasMonotoneAP
theorem Erdos195.erdos_195.variants.leq_5_bound Geneson [Ge19] proved that k ≤ 5 sorry line 53 source
Source documentation
Geneson [Ge19] proved that k ≤ 5.

depends on sorry what this means
Statement uses: GE.ge, Nat, OfNat.ofNat, SupSet.sSup, Set.ofPred, Equiv, Int, HasMonotoneAP
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_195.variants.leq_5_bound :
    5 ≥ sSup {k : ℕ | ∀ f : ℕ ≃ ℤ, HasMonotoneAP f k} := by
…
```
Mentions: HasMonotoneAP
theorem Erdos195.erdos_195.variants.leq_4_bound Adenwalla [Ad22] proved that k ≤ 4 sorry line 61 source
Source documentation
Adenwalla [Ad22] proved that k ≤ 4.

depends on sorry what this means
Statement uses: GE.ge, Nat, OfNat.ofNat, SupSet.sSup, Set.ofPred, Equiv, Int, HasMonotoneAP
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
16 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_195.variants.leq_4_bound :
    4 ≥ sSup {k : ℕ | ∀ f : ℕ ≃ ℤ, HasMonotoneAP f k} := by
…
```
Mentions: HasMonotoneAP