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Ben Green's Open Problem 21: green_21.variants.fox_kleitman_modular

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Added 1 Nov 2026 · not yet checked against the literature · importance 0.50

What the prompt says

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Statement

A question [FoKl06, Conjecture 5] of Fox and Kleitman, which they call a 'modular analogue' of Rado's Boundedness Conjecture. Let $k > 0$ , let $p$ be a prime, and suppose that $a_1, \dots, a_k$ are integers with $\sum_{i \in I} a_i \equiv 0 \pmod p$ only when $I = \emptyset$ . Does there exist an $f(k)$ -colouring of $(\mathbb{Z}/p\mathbb{Z})^*$ with no monochromatic solution to $a_1x_1 + \cdots + a_kx_k = 0$ ? This seems to be open even when $k = 3$ ; Green [Gr24] suspects the answer may be negative.

The point of the question is that the number of colours $f(k)$ must not depend on $p$ .

The arity $k$ is required to be positive, as in [FoKl06]: for $k = 0$ the empty tuple is a monochromatic solution with sum $0$ for every colouring, so no $f(0)$ could work.

Lean statement

theorem green_21.variants.fox_kleitman_modular : ∃ f : ℕ → ℕ,
    ∀ (k p : ℕ), 0 < k → p.Prime → ∀ a : Fin k → ℤ,
      (∀ I : Finset (Fin k), (p : ℤ) ∣ ∑ i ∈ I, a i → I = ∅) →
      ∃ col : (ZMod p)ˣ → Fin (f k), ∀ x : Fin k → (ZMod p)ˣ,
        (∀ i j, col (x i) = col (x j)) → ∑ i, (a i : ZMod p) * (x i : ZMod p) ≠ 0 := by
  sorry

Sources(1)

formal-conjectures: FormalConjectures/GreensOpenProblems/21.lean (opens in a new tab)

Openness evidence

Tagged category research open in google-deepmind/formal-conjectures at commit a64e959c32c2. Definitions used by the statement are in FormalConjectures/GreensOpenProblems/21.lean.

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