FormalConjectures.ErdosProblems.«844»

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import FormalConjecturesUtil

Erdős Problem 844

References:

erdosproblems.com/844

[AMS25] Alexeev, B., Mixon, D. and Sawin, W., The independence and clique cover numbers of the squarefree graph. arXiv:2507.01928 (2025).

[Ch74] Chvátal, V., Intersecting families of edges in hypergraphs having the hereditary property. (1974), 61-66.

namespace Erdos844

Those $n \in {1,\ldots,N}$ which are even or not squarefree, that is, the set of even numbers together with the odd non-squarefree numbers.

def evenOrOddNonSquarefree (N : ℕ) : Finset ℕ :=
  (Finset.Icc 1 N).filter (fun n => 2 ∣ n ∨ ¬ Squarefree n)

Let $A\subseteq {1,\ldots,N}$ be such that, for all $a,b\in A$, the product $ab$ is not squarefree.

Is the maximum size of such an $A$ achieved by taking $A$ to be the set of even numbers and odd non-squarefree numbers?

A problem of Erdős and Sárközy.

Weisenberg has provided the following positive proof. It is clear that such a maximal $A$ must contain all non-squarefree numbers. It therefore suffices to find the largest size of a subset of all squarefree numbers in ${1,\ldots,N}$ such that any two have at least one prime factor in common. By the result of Chvátal [Ch74] discussed in [701] this is maximised by the set of all even squarefree numbers.

An alternative proof was independently found by Alexeev, Mixon, and Sawin [AMS25].

This was formalized in Lean by Jennings using Aristotle.

@[category research solved, AMS 5 11, formal_proof using lean4 at
"https://gist.githubusercontent.com/JohnEdwardJennings/e32f2c412b0225091e7519d60741bd2d/raw/7d811ea413e2f7c0c0442749958aaac421eb6807/Erdos844.lean"]
theorem erdos_844 : answer(True) ↔ ∀ N : ℕ,
    IsGreatest {k : ℕ | ∃ A ⊆ Finset.Icc 1 N,
      (∀ a ∈ A, ∀ b ∈ A, ¬ Squarefree (a * b)) ∧ A.card = k}
      (evenOrOddNonSquarefree N).card := by⊢ True ↔
  ∀ (N : ℕ),
    IsGreatest {k | ∃ A ⊆ Finset.Icc 1 N, (∀ a ∈ A, ∀ b ∈ A, ¬Squarefree (a * b)) ∧ A.card = k}
      (evenOrOddNonSquarefree N).card
  sorryAll goals completed! 🐙

end Erdos844