Alphabeta MathThe Erdős–Hajnal Theorems for the E-Graph and Bird
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1 · Prerequisites
Abstract Algebra
Combinatorics
- Blockades, Combs and Pattern Graphs
- Bull-Free Graphs and the Erdős-Hajnal Property
- Cographs, Perfect Patterns and Pure Pairs
- Comb Structure in co-Bird-Free Graphs
- Comb Structure in co-E-Free Graphs
- Finite Counting, Factorials and Binomial Coefficients
- Finite Probability and the Probabilistic Method
- Finite Probability Spaces and Random Variables
- From Generalized Niceness to Erdős-Hajnal
- Generalized Niceness and Reduction Outcomes
- Graph Colouring
- Graphs, Walks and Connectivity
- Inclusion–Exclusion, the Pigeonhole Principle and Double Counting
- Induced Subgraphs and Hereditary Graph Classes
- Iterative Restriction and Comb-Extraction Lemmas
- Iterative Sparsification and the Five-Vertex Path
- Leaf Reducibility and Wonderful Families
- Modules, Substitution and Prime Graphs
- Polynomial Rödl, Virality and Erdős–Hajnal Equivalence
- Property (*) and Comb Outcomes
- Pure Pairs, Forests and Path–Antipath Classes
- Quotient Blockades and Mixing Relations
- Ramsey Theory
- Regular Pairs and Induced Counting
- Small-Graph Erdős-Hajnal Consequences
- Sparse Restricted Subgraphs and the Rödl–Nikiforov Theorems
- The Erdős–Hajnal Property and Homogeneous Sets
- The Five-Cycle and the Erdős-Hajnal Property
- The Structural Criterion for Property (*)
- Trees, Forests and Spanning Trees
Linear Algebra
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- 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
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Lemma
The E-graph and Bird singleton families are leaf-reducible
Corollary
The singleton E-graph family is generalized nice
Theorem
The E-graph has the Erdős-Hajnal property
Corollary
The singleton Bird family has property (*)
Corollary
The singleton Bird family is generalized nice
Theorem
The Bird graph has the Erdős-Hajnal property
Sources
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Section 2.1
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Lemmas 4.5, 5.1, 6.3 and 6.4
- Nguyen, Notes on Recent Work on the Erdős-Hajnal Conjecture, Section 5 iterative-sparsification context
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Theorem 1.10 and Section 6
- Nguyen, Notes on Recent Work on the Erdős-Hajnal Conjecture, Section 5, restricted-set/blockade exponent mechanism
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Lemmas 5.1 and 6.5, Section 6.2
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Lemma 4.5 and Section 6.2
- Shenwei Huang, Yiao Ju, and Yidong Zhou, Erdős-Hajnal beyond the five-vertex path, Theorem 1.11 and Section 6