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import FormalConjecturesUtilErdős Problem 484
References:
[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.
namespace Erdos484open scoped Classical inProve 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.
@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/1d7b3f00780b85ed0462e79a1cd5650ee9055655/src/v4.29.1/ErdosProblems/Erdos484.lean"]
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) := ⊢ ∃ c,
0 < c ∧
∀ (k : ℕ),
0 < k →
∃ N₀,
∀ (N : ℕ),
N₀ ≤ N →
∀ (f : ℕ → Fin k),
c * ↑N ≤
↑{n ∈ Finset.Icc 1 N | ∃ a ∈ Finset.Icc 1 N, ∃ b ∈ Finset.Icc 1 N, a ≠ b ∧ f a = f b ∧ a + b = n}.card
All goals completed! 🐙end Erdos484