FormalConjectures.ErdosProblems.«45»

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import FormalConjecturesUtil

Erdős Problem 45

References:

erdosproblems.com/45

[Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.

[Er96b] Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.

[Cr03] Croot, III, Ernest S., On a coloring conjecture about unit fractions. Ann. of Math. (2) (2003), 545-556.

[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

namespace Erdos45

Let $k\geq 2$. Is there an integer $n_k$ such that, if $D={ 1<d<n_k : d\mid n_k}$, then for any $k$-colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$?

This follows from the colouring result of Croot [Cr03]. Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$ (simply taking $n_k$ to be the lowest common multiple of some interval $[1,C^k]$). Sawhney has observed that there is also a doubly exponential lower bound, and hence this bound is essentially sharp.

@[category research solved, AMS 5 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos45.lean"]
theorem erdos_45 : answer(True) ↔
    ∀ k : ℕ, 2 ≤ k → ∃ n : ℕ, ∀ colouring : ℕ → Fin k,
      ∃ colour : Fin k, ∃ D' ⊆ {d ∈ n.divisors | 1 < d ∧ d < n},
        (∀ d ∈ D', colouring d = colour) ∧ D'.reciprocalSum = 1 := by⊢ True ↔
  ∀ (k : ℕ),
    2 ≤ k →
      ∃ n,
        ∀ (colouring : ℕ → Fin k),
          ∃ colour, ∃ D' ⊆ {d ∈ n.divisors | 1 < d ∧ d < n}, (∀ d ∈ D', colouring d = colour) ∧ D'.reciprocalSum = 1
  sorryAll goals completed! 🐙

end Erdos45