Alphabeta MathHow 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.
Every -sparse set of size contains a stable set of size at least
Statement
Every -sparse set of size contains a stable set of size at least .
Facts & Assumptions
Given: An integer divisible by , and a graph that is the disjoint union of four cliques, each of order .
The whole vertex set of this graph is -sparse (In a disjoint union of cliques of order the whole vertex set is -sparse, -sparse, -dense and -restricted vertex sets).
A stable set meets each clique in at most one vertex (Cliques, stable sets, the clique number and stability number , The connected components of a graph partition its vertex set and are its maximal connected subgraphs).
Refutation
By [L1], the whole vertex set of is a -sparse set of size .
By [L2], every stable set of has size at most , because there are only four clique components.
Since , one has . So the sparse set of step 1.1 contains no stable set of size at least half its order. Therefore the claim is false.
Depends on
- $c$-sparse, $c$-dense and $c$-restricted vertex sets
- In a disjoint union of cliques of order $k$ the whole vertex set is $(k/n)$-sparse
- Cliques, stable sets, the clique number $\omega(G)$ and stability number $\alpha(G)$
- The connected components of a graph partition its vertex set and are its maximal connected subgraphs
- Connected graphs and connected components defined by the existence of vertex paths
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Y. Huang, Q. Ju, and X. Zhou, Erdős-Hajnal beyond the five-vertex path, sec. 1.1 (standard reference, not scraped)