FormalConjectures.ErdosProblems.«202»

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import FormalConjecturesUtil

Erdős Problem 202

References:

erdosproblems.com/202

[BFV13] de la Bretèche, Régis and Ford, Kevin and Vandehey, Joseph, On non-intersecting arithmetic progressions. Acta Arith. (2013), 381-392.

[Ch05] Chen, Yong-Gao, On disjoint arithmetic progressions. Acta Arith. (2005), 143-148.

[Cr03b] Croot, III, Ernest S., On non-intersecting arithmetic progressions. Acta Arith. (2003), 233-238.

[ErSz68] Erdős, P. and Szemerédi, E., On a problem of P. Erdős and S. Stein. Acta Arith. (1968), 85-90.

[PaPh24] Park, Jinyoung and Pham, Huy Tuan, A proof of the Kahn-Kalai conjecture. J. Amer. Math. Soc. (2024), 235-243.

open Filter Realnamespace Erdos202

f N is the maximum possible r such that there are moduli $n_1<\cdots<n_r\leq N$ with associated residues $a_i\pmod{n_i}$ whose congruence classes are disjoint.

noncomputable def f (N : ℕ) : ℕ :=
  sSup {r : ℕ | ∃ n : Fin r → ℕ, ∃ a : Fin r → ℤ,
    StrictMono n ∧ (∀ i, 0 < n i ∧ n i ≤ N) ∧
    ∀ m : ℤ, ∀ i j : Fin r, m ≡ a i [ZMOD (n i : ℤ)] → m ≡ a j [ZMOD (n j : ℤ)] → i = j}

Let $n_1<\cdots < n_r\leq N$ with associated $a_i\pmod{n_i}$ such that the congruence classes are disjoint (that is, every integer is $\equiv a_i\pmod{n_i}$ for at most one $1\leq i\leq r$). How large can $r$ be in terms of $N$?

Let $f(N)$ be the maximum possible $r$, and let $L(N)=\exp(\sqrt{\log N\log\log N})$.

This was proved by GPT-5.4 Pro (prompted by Ho Boon Suan), using the argument of [BFV13] together with the resolution of the Kahn-Kalai conjecture by Park and Pham [PaPh24], so that $$f(N)= N L(N)^{-1+o(1)}.$$

@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos202.lean"]
theorem erdos_202 : ∃ o : ℕ → ℝ, o =o[atTop] (1 : ℕ → ℝ) ∧
    ∀ᶠ N : ℕ in atTop, (f N : ℝ) = (N : ℝ) * scaleL N ^ (-1 + o N) := by⊢ ∃ o, o =o[atTop] 1 ∧ ∀ᶠ (N : ℕ) in atTop, ↑(f N) = ↑N * scaleL N ^ (-1 + o N)
  sorryAll goals completed! 🐙

Erdős and Stein conjectured that $f(N)=o(N)$, which was proved by Erdős and Szemerédi [ErSz68].

@[category research solved, AMS 5 11]
theorem erdos_202.variants.erdos_szemeredi :
    (fun N => (f N : ℝ)) =o[atTop] (fun N : ℕ => (N : ℝ)) := by⊢ (fun N ↦ ↑(f N)) =o[atTop] fun N ↦ ↑N
  sorryAll goals completed! 🐙

end Erdos202