FormalConjectures.ErdosProblems.«160»

/-
Copyright 2025 The Formal Conjectures Authors.

Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at

    https://www.apache.org/licenses/LICENSE-2.0

Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
-/

import FormalConjecturesUtil

Erdős Problem 160

Reference: erdosproblems.com/160


namespace Erdos160

Let $h(n)$ be the smallest $k$ such that ${1,\ldots,n}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colours.

noncomputable def erdos_160.h (n : ℕ) : ℕ :=
    sInf {k | ∃ (colouring : Finset.Icc 1 n → Fin k), ∀ (progression : Set ℕ),
    (progression ⊆ Finset.Icc 1 n ∧ progression.IsAPOfLength 4) →
    3 ≤ (colouring '' {k | (k : ℕ) ∈ progression}).ncard}



open Filter

On Mathoverflow user leechlattice shows that $h(n) \ll n^{\frac 2 3}$.

@[category research solved, AMS 5 51]
theorem declaration uses 'sorry'erdos_160.known_upper :
    (fun n => (erdos_160.h n : ℝ)) =O[atTop] fun n => (n : ℝ) ^ ((2 : ℝ) / 3) := by⊢ (fun n => ↑(h n)) =O[atTop] fun n => ↑n ^ (2 / 3)
  sorryAll goals completed! 🐙


open Real

Estimate $h(n)$ by finding a better upper bound.

@[category research open, AMS 5 51]
theorem declaration uses 'sorry'erdos_160.better_upper :
    let upper_bound : ℕ → ℝ := answer(sorry)
    (fun n => (erdos_160.h n : ℝ)) =O[atTop] upper_bound ∧
    upper_bound =o[atTop] fun n => (n : ℝ) ^ ((2 : ℝ) / 3) := by⊢ let upper_bound := sorry;
(fun n => ↑(h n)) =O[atTop] upper_bound ∧ upper_bound =o[atTop] fun n => ↑n ^ (2 / 3)
  sorryAll goals completed! 🐙

Estimate $h(n)$ by finding a better lower bound.

@[category research open, AMS 5 51]
theorem declaration uses 'sorry'erdos_160.better_lower:
    let lower_bound : ℕ → ℝ := answer(sorry)
    (lower_bound =O[atTop] fun n => (erdos_160.h n : ℝ)) ∧
    ∀ c > 0,
    (fun (n : ℕ) => exp (c * log n ^ ((1 : ℝ) / 12))) =O[atTop] (fun n => (erdos_160.h n : ℝ)) →
    ∀ c > 0, (fun (n : ℕ) => exp (c * log n ^ ((1 : ℝ) / 12))) =o[atTop] lower_bound := by⊢ let lower_bound := sorry;
(lower_bound =O[atTop] fun n => ↑(h n)) ∧
  ∀ c > 0,
    ((fun n => rexp (c * log ↑n ^ (1 / 12))) =O[atTop] fun n => ↑(h n)) →
      ∀ c > 0, (fun n => rexp (c * log ↑n ^ (1 / 12))) =o[atTop] lower_bound
  sorryAll goals completed! 🐙

The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka KeMe23 imply that $$h(N) \gg \exp(c(\log N)^{\frac 1 {12}})$$ for some $c > 0$.

@[category research solved, AMS 5 51]
theorem declaration uses 'sorry'erdos_160.variants.known_lower :
    ∃ c > 0, (fun (n : ℕ) => exp (c * log (n : ℝ) ^ ((1 : ℝ) / 12)))
    =O[atTop] fun n => (erdos_160.h n : ℝ):= by⊢ ∃ c > 0, (fun n => rexp (c * log ↑n ^ (1 / 12))) =O[atTop] fun n => ↑(h n)
  sorryAll goals completed! 🐙


end Erdos160