DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Tournament property $S_{k}$ : every set of at most $k$ vertices is dominated by one vertex

Definition

For $k \in N$ , a tournament $T = (V, A)$ has property $S_{k}$ when for every set $S \subseteq V$ with $∣ S ∣ \leq k$ , there is a vertex $v \in V ∖ S$ such that $(v, s) \in A$ for every $s \in S$ . For $S = \emptyset$ , this requires $V$ to be nonempty.

Depends on

A tournament is an orientation of a complete finite graph

Used by

If k≥1 and n≥3k² 2ᵏ, an n-vertex tournament with property Sₖ exists Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 results over 6 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

M. Bucic, Probabilistic Method, Section 1.1 (standard reference, not scraped)

Tournament property SkSk​: every set of at most kk vertices is dominated by one vertex