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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Szele's bound: for every $n \geq 1$ , some $n$ -vertex tournament has at least $n! / 2^{n - 1}$ Hamilton paths

Statement

For every natural $n \geq 1$ , some tournament on $n$ labelled vertices has at least $\frac{n!}{2^{n - 1}}$ directed Hamilton paths.

Facts & Assumptions

Given: A labelled vertex set $V$ of size $n \geq 1$ .

[L1]

A tournament orients exactly one direction between each two distinct vertices (A tournament is an orientation of a complete finite graph).

[L2]

A directed path is a directed walk $v_{0}, \dots, v_{ℓ}$ , with $(v_{i - 1}, v_{i})$ an arc at every step, whose vertices are distinct (Directed walks, trails, paths and cycles, and strong connectivity). A directed Hamilton path is one containing every vertex.

[L3]

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

[L4]

A finite $n$ -element set has exactly $n!$ bijective orderings (A finite set $A$ with $∣ A ∣ = n$ has exactly $n!$ bijections onto itself, and $n!$ bijections onto any set of the same cardinality).

[L5]

Indicators count events, expectation is linear, and some outcome reaches at least its expectation (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, Expectation preserves pointwise order and lies between the minimum and maximum attained values).

Proof

technique · direct

1.1

Orient every possible edge independently and fairly. A fixed ordering of the vertices is a directed Hamilton path exactly when its $n - 1$ consecutive edges receive prescribed orientations, an event of probability $2^{- (n - 1)}$ .

L1L2L3

2.1

Sum an indicator over the $n!$ orderings. Its expectation is $n! / 2^{n - 1}$ .

step 1.1L4L5

3.1

Some tournament has at least this many directed Hamilton paths. For $n = 1$ , the unique ordering is a Hamilton path and the bound is $1$ .

step 2.1L5∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 81 results over 17 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 3.2.1 (standard reference, not scraped)
M. Bucic, Probabilistic Method, Theorem 2.2 (standard reference, not scraped)

Szele's bound: for every n≥1n≥1, some nn-vertex tournament has at least n!/2n−1n!/2n−1 Hamilton paths