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Pure Pairs, Forests and Path–Antipath Classes — Examples
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Finite Counting, Factorials and Binomial Coefficients
- Finite Probability Spaces and Random Variables
- Foundations of the Real Numbers for Analysis
- Graphs, Walks and Connectivity
- Inclusion–Exclusion, the Pigeonhole Principle and Double Counting
- Induced Subgraphs and Hereditary Graph Classes
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Metric Spaces
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Order, Zorn's Lemma, and the Axiom of Choice
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Pure Pairs, Forests and Path–Antipath Classes
- Regular Pairs and Induced Counting
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Sparse Restricted Subgraphs and the Rödl–Nikiforov Theorems
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Erdős–Hajnal Property and Homogeneous Sets
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The ZFC Axioms and the Basic Set Constructions
- Topology of ℝ
- Trees, Forests and Spanning Trees
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Summary
These examples record the first concrete endpoint of the page's path–antipath theorem and make the co-leaf convention explicit on the standard test case. The remaining planned examples depend on forest and leaf-deletion results that are still blocked at step 5.
3 · Logical flowchart
4 · Definitions, theorems and proofs
None yet.
5 · Examples, counterexamples and false statements
The path-antipath theorem specialized to and
Example
Graphs with no induced and no induced have the Erdős–Hajnal property.
Facts & Assumptions
Given: The standard five-vertex path .
For every integer , the class forbidding and has the Erdős–Hajnal property (For every , the class forbidding and has the Erdős–Hajnal property).
The graph is the standard path on five vertices (Empty and complete graphs, complete bipartite graphs, and the convention that and have vertices).
Verification
By [L2], the parameter choice in [L1] is exactly the class forbidding and .
Therefore graphs with no induced and no induced have the Erdős–Hajnal property.
A co-leaf of is exactly a leaf of
Example
Let have vertices in path order. In the vertices and are co-leaves, and no other vertex is.
Facts & Assumptions
Given: The five-vertex path with edges .
A co-leaf of a graph is a vertex that is a leaf in the complement, equivalently a vertex of degree (Co-leaves of a finite graph).
The complement contains exactly the nonedges of the original graph (Graph isomorphisms, automorphisms and graph complements).
Verification
By [L2], the complement has edges . So vertices and each have degree in , while vertices each have degree .
Step 1.1 and [L1] show that and are co-leaves of , and that are not. Since and are exactly the leaves of the path , the co-leaves of are precisely the leaves of .