The Manickam–Miklós–Singhi conjecture (1988)

The Manickam–Miklós–Singhi conjecture (1988). If n ≥ 4k and x_1, …, x_n are real numbers with x_1 + … + x_n = 0, then at least C(n-1, k-1) of the k-element subsets S ⊆ 1, …, n have Σ_i ∈ S x_i ≥ 0.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-paper-manickam-miklos-singhi-conjecture,
  title        = {The Manickam–Miklós–Singhi conjecture (1988)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/paper-manickam-miklos-singhi-conjecture}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

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Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

The Manickam–Miklós–Singhi conjecture (1988).

If $n \geq 4 k$ and $x_{1}, \dots, x_{n}$ are real numbers with $x_{1} + \dots + x_{n} = 0$ , then at least $(k - 1 n - 1)$ of the $k$ -element subsets $S \subseteq {1, \dots, n}$ have $\sum_{i \in S} x_{i} \geq 0$ . (The bound is attained by $x = (n - 1, - 1, \dots, - 1)$ ; the condition $n \geq 4 k$ cannot be weakened to $n \geq 3 k + 1$ in general.)

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.Paper.ManickamMiklosSinghiConjecture.

theorem manickam_miklos_singhi_conjecture :
    ∀ (n k : ℕ), 1 ≤ k → 4 * k ≤ n → ∀ x : Fin n → ℝ, ∑ i, x i = 0 →
      (n - 1).choose (k - 1) ≤ nonnegSubsetCount n k x

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
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A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

[MM88] Manickam, N. and Miklós, D. (1988). "On the number of nonnegative partial sums of a nonnegative sum." Colloq. Math. Soc. János Bolyai 52, pp. 385--392.
[MS88] Manickam, N. and Singhi, N. M. (1988). "First distribution invariants and EKR theorems." J. Combin. Theory Ser. A 48, pp. 91--103.
[CSS14] Chowdhury, A., Sarkis, G. and Shahriari, S. (2014). "The Manickam–Miklós–Singhi conjectures for sets and vector spaces." J. Combin. Theory Ser. A 128, pp. 84--103. arXiv:1309.2212
[Po15] Pokrovskiy, A. (2015). "A linear bound on the Manickam–Miklós–Singhi conjecture." J. Combin. Theory Ser. A 133, pp. 280--306. arXiv:1308.2176

Source and licence

Imported from Formal Conjectures (research papers), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.