LemmaStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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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.

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.

A prescribed set of present and absent edges in $G (n, p)$ has product probability

Statement

In $G (n, p)$ , let $R$ and $F$ be disjoint sets of possible edges, with $∣ R ∣ = r$ and $∣ F ∣ = s$ . The probability that every edge of $R$ is present and every edge of $F$ is absent is $p^{r} (1 - p)^{s} .$ In particular, a fixed labelled graph with $m$ edges has probability $p^{m} (1 - p)^{(\binom{n}{2}) - m}$ .

Facts & Assumptions

Given: $G (n, p)$ and disjoint prescribed edge sets $R, F$ .

[L1]

Coordinate events in a finite product probability space are mutually independent (Product weights normalize, and coordinate events are mutually independent).

[L2]

$G (n, p)$ has independent Bernoulli $(p)$ coordinates indexed by the possible edges (The Erdős-Rényi finite random graph $G (n, p)$ ).

[L3]

There are $(\binom{n}{2})$ possible edges on $[n]$ (The set $[A]^{k}$ of $k$ -element subsets and the binomial coefficient $(\binom{n}{k}) : = ∣ [n]^{k} ∣$ ).

Proof

technique · direct

1.1

Each required-present coordinate has probability $p$ and each required-absent coordinate has probability $1 - p$ .

2.1

Mutual independence factors the joint probability as $p^{r} (1 - p)^{s}$ . This remains valid for empty prescriptions and for $p = 0, 1$ .

step 1.1L1algebra

3.1

For a fixed graph, take its $m$ edges as $R$ and the remaining $(\binom{n}{2}) - m$ possible edges as $F$ .

step 2.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 62 results over 20 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

J. Matousek and J. Vondrak, The Probabilistic Method, Section 1.1 (standard reference, not scraped)

A prescribed set of present and absent edges in G(n,p)G(n,p) has product probability