1. FormalConjectures
  2. GreensOpenProblems
  3. «4»
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Ben Green's Open Problem 4

Reference: [Ben Green's Open Problem 4](https://people.maths.ox.ac.uk/greenbj/papers/open-problems.pdf#section.4 Problem 4)

namespace Green4

A set in a monoid is product-free if there are no elements x, y, z in the set such that x * y = z.

def ProdFree {M : Type*} [Monoid M] (S : Set M) : Prop := ∀ x ∈ S, ∀ y ∈ S, x * y ∉ S

What is the largest product-free set in the alternating group $A_n$?

@[category research open, AMS 20] theorem declaration uses 'sorry'green_4 (n : ℕ) : let S : ∀ n, Set (alternatingGroup <| Fin n) := answer(sorry) MaximalFor (ProdFree (M := alternatingGroup <| Fin n)) Set.ncard (S n) := n:ℕ⊢ let S := sorry; MaximalFor ProdFree Set.ncard (S n) All goals completed! 🐙

Defines a family of subsets of $A_n$ where each permutation $\pi$ in a subset obeys $\pi(x)$ and $\forall v \in I$, \pi(v)\notin I$ for a fixed $x$ and $I$. It is easy to demonstrate that such a subset is product-free, because for any a,b,c in such a set, $(a*b) (x)=a(b(x))\notin I$ but $c(x) in I$

def extremalFamily {n : ℕ} (x : Fin n) (I : Set (Fin n)) : Set <| alternatingGroup <| Fin n := {π | π.val x ∈ I ∧ Disjoint (I.image π) I}

In the case of large n, the problem was solved in On the largest product-free subsets of the alternating groups. Specifically, this theorem formalizes the statement of theorem 1.1 in the mentioned paper

@[category research solved, AMS 20] theorem declaration uses 'sorry'large_green_4 : ∀ᶠ n in .atTop, ∀ S, MaximalFor (ProdFree (M := alternatingGroup <| Fin n)) Set.ncard S → ∃ x I, S = extremalFamily x I ∨ S = (extremalFamily x I).image (·⁻¹) := ⊢ ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (S : Set ↥(alternatingGroup (Fin n))), MaximalFor ProdFree Set.ncard S → ∃ x I, S = extremalFamily x I ∨ S = (fun x => x⁻¹) '' extremalFamily x I All goals completed! 🐙 end Green4