Alphabeta MathQuantitative Induced Density and the Log-Log Step
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
1 · Prerequisites
- Absolute and Conditional Convergence; Rearrangement; Products
- Binary Operations, Monoids, Groups and Subgroups
- Blockades, Combs and Pattern Graphs
- 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
- 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
- Regular Pairs and Induced Counting
- Relations, Functions, and Quotients
- 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 Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The ZFC Axioms and the Basic Set Constructions
- Topology of ℝ
- Vector Spaces, Linear Subspaces, Span and Direct Sums
2 · Logical flowchart
3 · Items
Definition
Induced copy density and homogeneous restriction parameter
Definition
Qid restricted blockade with empty blocks
Definition
Labelled blowup and good induced copy
Lemma
Good copy extension count
Lemma
Few induced copies exclude a fixed labelled blowup
Lemma
Qid bipartite density trimming
Lemma
Qid fixed size density selection
Lemma
Local special copy trichotomy
Lemma
Qid maximal blowup trichotomy
Lemma
Special copy trichotomy produces a restricted blockade
Definition
Subreciprocal function and ell divisibility
Lemma
Admissible parameters for the density recursion
Definition
Qid finite density recursion profile
Lemma
Qid logarithmic and constant divisibility
Lemma
Ell divisibility amplifies through a blockade
Theorem
Quantitative density theorem for ell divisive graphs
Corollary
Fox sudakov quantitative induced density bound
Theorem