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CorollaryStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28
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Every perfect graph has a clique or stable set of size at least the square root of its order

Statement

If G is a perfect graph on n vertices, then G has a clique or a stable set of size at least n.

Facts & Assumptions

Given: A perfect graph G on n vertices.

[L1]

Perfect graphs satisfy ∣V(G)∣≤κ(G) (Every perfect graph satisfies |V(G)|<=kappa(G)).

[L2]

κ(G)=α(G)ω(G) and hom⁡(G)=max⁡{α(G),ω(G)} (The parameter kappa(G)=alpha(G)omega(G), Homogeneous vertex sets and the homogeneous number hom⁡(G)=max⁡{ω(G),α(G)}, Cliques, stable sets, the clique number ω(G) and stability number α(G)).

Proof

technique · direct
1.1

Let h:=hom⁡(G)=max⁡{α(G),ω(G)}. Then α(G)≤h and ω(G)≤h, so [L2] gives κ(G)=α(G)ω(G)≤h2.

L2given
2.1

Since n=∣V(G)∣, [L1] and step 1.1 yield n≤h2. Therefore h≥n. If h=ω(G), G has a clique of size at least n; if h=α(G), it has a stable set of that size.

step 1.1L1L2
3.1

Hence every perfect graph on n vertices has a clique or stable set of size at least n.

step 2.1∎

Depends on

Used by

Dependency tree · two levels

14 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.

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