Source labels openChecked July 26, 2026

Beck–Fiala theorem and conjecture

The Beck–Fiala conjecture*

Mathematical statement

The Beck–Fiala conjecture*

There exists a universal constant $C > 0$ such that every set system $S_1, \dots, S_m \subseteq [n]$ of degree at most $t$ admits a colouring $\chi \colon [n] \to \{-1, +1\}$ with $\left|\sum_{j \in S_i} \chi(j)\right| \le C \sqrt{t}$ for every $i$ .

Statement source: Wikipedia statement material

Statement terms: CC-BY-SA-4.0

Attributed source material. Reuse must follow the linked attribution and share-alike terms.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

beck_fiala_conjecture

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem beck_fiala_conjecture :    ∃ C : ℝ, 0 < C ∧ ∀ (n m t : ℕ) (S : Fin m → Finset (Fin n)),      (∀ j, (Finset.univ.filter fun i => j ∈ S i).card ≤ t) →      ∃ χ : Fin n → ℝ, (∀ j, χ j = 1 ∨ χ j = -1) ∧        ∀ i, |∑ j ∈ S i, χ j| ≤ C * Real.sqrt t := by  sorry

Statement source: Formal Conjectures
Lean version: v4.27.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Pinned Formal Conjectures sourceformalization Wikipediabackground J. Beck and T. Fiala, *"Integer-making" theorems*, Discrete Applied Mathematics **3** (1981), 1–8background B. Bukh, *An improvement of the Beck–Fiala theorem*, Combinatorics, Probability and Computing **25** (2016), 380–398background