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The singleton family has property (*)
Statement
The singleton finite family has property .
Facts & Assumptions
Given: An arbitrary co--free graph and a special-vertex comb required by property .
has the Erdős–Hajnal property (The family consisting of and co- has the Erdős–Hajnal property).
The special-vertex comb has the required partition (A special-vertex comb in a co--free graph admits the structural partition).
The local criterion converts those two facts into property (The special-vertex-local structural-partition criterion implies property (*)).
Proof
For , its complement family is . Thus the given graph is in the setting of [F2].
Take . Fact [F1] supplies their common Erdős–Hajnal constant, and [F2] supplies the local partition for every special-vertex comb in the graph of step 1.1.
Applying [F3] now proves that has property .
Depends on
- The special-vertex-local structural-partition criterion implies property (*)
- The family consisting of $H_5$ and co-$E$ has the Erdős–Hajnal property
- A special-vertex comb in a co-$E$-free graph admits the $\{H_5,\mathrm{co}\text{-}E\}$ structural partition
- Property (*) for a finite graph family
- The $E$-graph and co-$E$
- $H$-free and $\mathcal F$-free graphs under the induced-subgraph convention
- Graph isomorphisms, automorphisms and graph complements
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
34 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
- Huang, Ju, and Zhou, Erdős-Hajnal beyond the five-vertex path, Sections 5--6.1 (standard reference, not scraped)