TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Every $k$ -uniform hypergraph with fewer than $2^{k - 1}$ edges is $2$ -colourable

Statement

Let $k \geq 1$ . Every finite $k$ -uniform hypergraph with fewer than $2^{k - 1}$ edges admits a vertex two-colouring with no monochromatic edge.

Facts & Assumptions

Given: A finite $k$ -uniform hypergraph $H = (V, E)$ with $k \geq 1$ and $∣ E ∣ < 2^{k - 1}$ .

[L1]

An edge in a $k$ -uniform hypergraph has exactly $k$ vertices ( $r$ -uniform hypergraphs and complete balanced $r$ -partite $r$ -graphs $K_{s, \dots, s}^{(r)}$ ).

[L2]

Independent coordinate events in a finite product space have product probability (Product weights normalize, and coordinate events are mutually independent).

[L3]

A sum of indicators counts the corresponding events, each indicator has expectation equal to its event probability, and expectation is linear without independence (Indicators turn event probabilities, intersections, and finite counts into expectations and products, Expectation is linear for every finite family of random variables, without any independence hypothesis).

[L4]

A nonnegative integer-valued variable with expectation less than $1$ vanishes at some outcome (The first-moment method for avoiding or forcing a finite count of bad events).

Proof

technique · direct

1.1

Colour every vertex independently and uniformly red or blue. For a fixed edge, its $k$ colours are all red or all blue with probability $2 \cdot 2^{- k} = 2^{1 - k}$ .

L1L2

2.1

Let $X$ count monochromatic edges. Then $E [X] = ∣ E ∣ 2^{1 - k} < 1$ .

step 1.1L3algebra

3.1

By [L4], some colouring has $X = 0$ and is proper. If $k = 1$ , the edge hypothesis forces $E = \emptyset$ , and the same proof applies.

step 2.1L4∎

Depends on

Used by

Checking the symmetric local-lemma condition for a sparse uniform hypergraph Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 54 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 2.2.4 (standard reference, not scraped)

Every kk-uniform hypergraph with fewer than 2k−12k−1 edges is 22-colourable