Alphabeta MathFinite Probability and the Probabilistic Method
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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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✓ 20 results · all verified · 11 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 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Linear Algebra
Number Theory
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
Theorem
An event of positive probability in a finite probability space is nonempty
Theorem
The first-moment method for avoiding or forcing a finite count of bad events
Proposition
The deletion-alteration method converts an expected defect count into a deterministic lower bound
Lemma
1+x≤exp(x) for every real x, hence (1-p)ᵐ≤exp(-mp)
Definition
The Erdős-Rényi finite random graph G(n,p)
Lemma
A prescribed set of present and absent edges in G(n,p) has product probability
Definition
The moment generating function M_X(t)=E[e^tX] on a finite probability space
Lemma
For a uniform random sign ε, E[e^tε]≤ e^t²/2
Theorem
For n≥1 independent random signs, ℙ(|S|≥ t)≤2 exp(-t²/(2n)) for t>0
Definition
Dependency digraphs for a finite family of bad events
Lemma
The conditional-probability induction underlying the Lovász Local Lemma
Theorem
The asymmetric Lovász Local Lemma for finitely many events
Corollary
The symmetric Lovász Local Lemma under ep(d+1)≤1
Theorem
Every k-uniform hypergraph with fewer than 2ᵏ⁻¹ edges is 2-colourable
Theorem
A k-uniform hypergraph is 2-colourable when every edge meets at most d other edges and e(d+1)≤2ᵏ⁻¹
Theorem
Every finite graph with m edges has a cut containing at least m/2 edges
Theorem
Szele's bound: for every n≥1, some n-vertex tournament has at least n!/2ⁿ⁻¹ Hamilton paths
Definition
Tournament property Sₖ: every set of at most k vertices is dominated by one vertex
Theorem
If k≥1 and n≥3k² 2ᵏ, an n-vertex tournament with property Sₖ exists
Definition
Dominating sets in a finite graph
Theorem
An n-vertex graph of minimum degree δ>1 has a dominating set of size at most n(log(δ+1)+1)/(δ+1)
Definition
Sum-free subsets of the integers
Lemma
There are arbitrarily large primes congruent to 2 modulo 3
Theorem
Every nonempty finite set of n nonzero integers has a sum-free subset of size greater than n/3
Lemma
The expected number of cycles of length at most ℓ in G(n,p)
Lemma
ℙ(α(G(n,p))≥ s)leC(n, s)(1-p)^C(s, 2)≤ nˢ exp(-pC(s, 2)) for s≤ n
Theorem
For all positive k,ℓ, some finite graph has girth greater than ℓ and chromatic number greater than k
Sources
- J. Matousek and J. Vondrak, The Probabilistic Method, Chapter 2
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Section 2.1
- J. Matousek and J. Vondrak, The Probabilistic Method, Chapter 4
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Section 6.3
- J. Matousek and J. Vondrak, The Probabilistic Method, Definition 1.1.2
- J. Matousek and J. Vondrak, The Probabilistic Method, Section 1.1
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Section 4.1
- J. Matousek and J. Vondrak, The Probabilistic Method, Section 7.1
- J. Matousek and J. Vondrak, The Probabilistic Method, Section 5.1
- J. Matousek and J. Vondrak, The Probabilistic Method, proof of Theorem 5.1.1
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, proof of Theorem 5.1
- J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 5.1.1
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Theorem 5.1
- J. Matousek and J. Vondrak, The Probabilistic Method, Corollary 5.1.2
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Corollary 5.3
- J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 2.2.4
- J. Matousek and J. Vondrak, The Probabilistic Method, Section 5.2
- J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 3.3.1
- M. Bucic, Probabilistic Method, Theorem 2.1
- J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 3.2.1
- M. Bucic, Probabilistic Method, Theorem 2.2
- M. Bucic, Probabilistic Method, Section 1.1
- M. Bucic, Probabilistic Method, Theorem 1.5
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Section 6.1
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Theorem 6.1
- M. Bucic, Probabilistic Method, Section 2.3
- M. Bucic, Probabilistic Method, proof of Theorem 2.3
- M. Bucic, Probabilistic Method, Theorem 2.3
- J. Matousek and J. Vondrak, The Probabilistic Method, proof of Theorem 4.2.1
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, proof of Theorem 6.3
- J. Matousek and J. Vondrak, The Probabilistic Method, Theorem 4.2.1
- Y. Zhao, MIT 18.218 Probabilistic Method in Combinatorics, Theorem 6.3
- M. Bucic, Probabilistic Method, Theorem 4.1