FormalConjectures.ErdosProblems.«155»

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import FormalConjecturesUtil

Erdős Problem 155

Reference: erdosproblems.com/155


open Filter


namespace Erdos155

Let $F(N)$ be the size of the largest Sidon subset of ${1, \dots, N}$.

noncomputable abbrev F (N : ℕ) : ℕ := Finset.maxSidonSubsetCard (Finset.Icc 1 N)

Is it true that for every $k \geq 1$ we have $$ F(N + k) \leq F(N) + 1 $$ for all sufficiently large $N$?

@[category research open, AMS 5]
theorem declaration uses 'sorry'erdos_155 : answer(sorry) ↔ ∀ k ≥ 1, ∀ᶠ N in atTop, F (N + k) ≤ F N + 1 := by⊢ True ↔ ∀ k ≥ 1, ∀ᶠ (N : ℕ) in atTop, F (N + k) ≤ F N + 1
  sorryAll goals completed! 🐙


-- TODO: This may even hold with $k \approx ε * N ^ (1 / 2)$.

end Erdos155