Alphabeta MathThe Erdős–Hajnal Property and Homogeneous Sets — Examples
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✓ 8 results · all verified · 7 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Combinatorics
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
Example
Kₙ and K̄ₙ both have homogeneous number n
Example
For positive a,b, hom(K_a,b)=max{2,a,b}
Example
The self-complementary five-cycle satisfies hom(C₅)=2
Example
The classes of complete graphs and of empty graphs have Erdős–Hajnal constant 1
Example
Every hereditary graph class of bounded order has the Erdős–Hajnal property
Example
Forbidding P₃ and forbidding P₃ have the same Erdős–Hajnal constants
Counterexample
The universal logarithmic Ramsey guarantee cannot be replaced by any universal positive power
False statement