Alphabeta MathFinite Probability and the Probabilistic Method — Examples and Counterexamples
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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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✓ 6 results · all verified · 2 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
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
The expected number of triangles in G(n,p) is C(n, 3)p³
Example
First- and second-moment bounds for a nonempty Bernoulli random subset
Example
The random-colouring proof of R(k,k)>2^k/2
Example
Checking the symmetric local-lemma condition for a sparse uniform hypergraph
Example
A parameter ledger for the high-girth, high-chromatic alteration proof
Counterexample
Expectation equal to 1 does not force a nonnegative integer-valued variable to vanish somewhere
Sources
- J. Matousek and J. Vondrak, The Probabilistic Method, Chapters 1 and 3
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Chapter 3
- J. Matousek and J. Vondrak, The Probabilistic Method, Section 2.1
- J. Matousek and J. Vondrak, The Probabilistic Method, Sections 5.1-5.2
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, proof of Theorem 6.3
- J. Matousek and J. Vondrak, The Probabilistic Method, Chapter 2