Alphabeta MathHow 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.
The Erdős-Rényi finite random graph
Definition
Let and . The Erdős-Rényi random graph is the finite simple graph on the labelled vertex set in which the possible edge indicators are mutually independent Bernoulli variables (Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials). Equivalently, its probability space is the product of one Bernoulli edge space for every two-element subset of .
Depends on
- The finite product of finite probability spaces
- Bernoulli random variables and binomial random variables as sums of independent Bernoulli trials
- A finite simple graph is a finite vertex set together with a set of two-element vertex subsets
- The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$
Used by
- The expected number of triangles in G(n,p) is C(n, 3)p³ Example
- The random-colouring proof of R(k,k)>2^k/2 Example
- A prescribed set of present and absent edges in G(n,p) has product probability Lemma
- The expected number of cycles of length at most ℓ in G(n,p) Lemma
- For every n≥16 there is an n-vertex graph with hom(G)<3 log₂ n Theorem
Dependency tree · two levels
18 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- J. Matousek and J. Vondrak, The Probabilistic Method, Definition 1.1.2 (standard reference, not scraped)