484.lean`

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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 484

References:

erdosproblems.com/484
[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[ESS89] Erdős, P., Sárközy, A., and Sós, V. T., On a conjecture of Roth and some related problems. I. (1989), 47-59.

Imports (1)

FormalConjecturesUtil

theorem Erdos484.erdos_484 Prove that there exists an absolute constant sorry line 48 source
Source documentation
Prove that there exists an absolute constant $c>0$ such that, whenever $\{1,\ldots,N\}$ is $k$-coloured (and $N$ is large enough depending on $k$) then there are at least $cN$ many integers in $\{1,\ldots,N\}$ which are representable as a monochromatic sum (that is, $a+b$ where $a,b\in \{1,\ldots,N\}$ are in the same colour class and $a\neq b$).
A conjecture of Roth. Solved by Erdős, Sárközy, and Sós [ESS89], who in fact prove that there are at least $\frac{N}{2}-O(N^{1-1/2^{k+1}})$ many even numbers which are of this form.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Nat, LE.le, Fin, HMul.hMul, Finset.card, Finset.filter, Membership.mem, Finset, Finset.Icc, Ne, Eq, HAdd.hAdd, Classical.propDecidable
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_484 :
    ∃ c : ℝ, 0 < c ∧ ∀ k : ℕ, 0 < k → ∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N → ∀ f : ℕ → Fin k,
      c * N ≤ (((Finset.Icc 1 N).filter fun n =>
        ∃ a ∈ Finset.Icc 1 N, ∃ b ∈ Finset.Icc 1 N,
          a ≠ b ∧ f a = f b ∧ a + b = n).card : ℝ) := by
…
```
Mentions: c