FormalConjectures.ErdosProblems.«34»

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import FormalConjecturesUtil

Erdős Problem 34

References:

erdosproblems.com/34

[Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.

[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

[He86] Hegyvári, N., On consecutive sums in sequences. Acta Math. Hungar. (1986), 193--200.

[Ko15] Konieczny, J., On consecutive sums in permutations. arXiv:1504.07156 (2015).


namespace Erdos34


open scoped BigOperators

The set of consecutive sums of a permutation p of {1,…,n}, where position k holds the value p k + 1: the sums ∑_{u ≤ k ≤ v} (p k + 1).

def consecutiveSums (n : ℕ) (p : Equiv.Perm (Fin n)) : Finset ℕ :=
  (Finset.univ.filter (fun x : Fin n × Fin n => x.1 ≤ x.2)).image
    (fun x => ∑ k ∈ Finset.Icc x.1 x.2, ((p k : ℕ) + 1))

For any permutation $\pi\in S_n$ of ${1,\ldots,n}$ let $S(\pi)$ count the number of distinct consecutive sums, that is, sums of the shape $\sum_{u\leq i\leq v}\pi(i)$. Is it true that $$ S(\pi) = o(n^2) $$ for all $\pi\in S_n$?

Hegyvári [He86] gave a counterexample.

@[category research solved, AMS 5 11,
  formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos34.lean"]
theorem declaration uses 'sorry'erdos_34 : answer(False) ↔
    ∀ c : ℝ, 0 < c → ∃ N : ℕ, ∀ n ≥ N, ∀ p : Equiv.Perm (Fin n),
      ((consecutiveSums n p).card : ℝ) < c * (n : ℝ) ^ 2 := by⊢ False ↔ ∀ (c : ℝ), 0 < c → ∃ N, ∀ n ≥ N, ∀ (p : Equiv.Perm (Fin n)), ↑(consecutiveSums n p).card < c * ↑n ^ 2
  sorryAll goals completed! 🐙


end Erdos34