188.lean`

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

4 declarations, 3 contain sorry, 0 declared as axioms

Erdős Problem 188

References:

erdosproblems.com/188
[EGMRSS75] Erdős, P. and Graham, R. L. and Montgomery, P. and Rothschild, B. L. and Spencer, J. and Straus, E. G., Euclidean {R}amsey theorems. {II}. (1975), 529--557.
[Ts17] Tsaturian, Sergei, A {E}uclidean {R}amsey result in the plane. Electron. J. Combin. (2017), Paper No. 4.35, 9.

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100OpenSet1.lean

def Erdos188.s The set of numbers line 39 source
Source documentation
The set of numbers $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance 1.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Set, Nat
Proof uses: Set.ofPred, Exists, Complex, And, Set.Pairwise, SDiff.sdiff, Set.univ, Ne, Real, Dist.dist, OfNat.ofNat, Not, LE.le, Eq, Norm.norm, Set.IsAPOfLengthWith, ENat
14 instances, coercions and auxiliary constants left out.
```
def s := { k : ℕ | ∃ blue : Set ℂ,
…
```
theorem Erdos188.erdos_188 What is the smallest sorry line 48 source
Source documentation
What is the smallest $k$ such that $\mathbb{R}^2$ can be red/blue coloured with no pair of red points unit distance apart, and no $k$-term arithmetic progression of blue points with distance 1?

depends on sorry what this means
Statement uses: IsLeast, Nat, Erdos188.s, sorryAx, Bool.false
Proof uses: Lean.Name, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_188 : IsLeast s answer(sorry) := by
…
```
Mentions: Erdos188.s
theorem Erdos188.erdos_188.variants.nonempty Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 14, 15): It has been shown that there is a large sorry line 59 source
Source documentation
Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 14, 15):
It has been shown that there is a large $M$ so that it is possible to partition $\mathbb{E}^2$ into two sets $A$ and $B$ so that $A$ contains no pair of points with distance 1 and $B$ contains no A.P. of length $M$.

depends on sorry what this means
Statement uses: Set.Nonempty, Nat, Erdos188.s
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
1 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_188.variants.nonempty : s.Nonempty := by
…
```
theorem Erdos188.erdos_188.variants.estimate Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 15): How small can $M$ be made? sorry line 69 source
Source documentation
Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 15):
How small can $M$ be made? The only estimate currently known is that $M \le 10000000$ (more or less). In the other direction, it has just been shown by R. Juhász [Ju (79)] that we must have $M \ge 5$.

depends on sorry what this means
Statement uses: And, Nat, Membership.mem, Set, Erdos188.s, LE.le, OfNat.ofNat, Exists
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
6 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_188.variants.estimate : (∀ k, k ∈ s → 5 ≤ k) ∧ (∃ k ∈ s, k ≤ 10000000) := by
…
```
Mentions: Erdos188.s