LemmaStatement: AI-generatedProof: AI-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.

Two fixed-parameter expander events have vanishing success probabilities

Statement

Let $n \geq 3$ be an integer and set $L = \log n$ , where $\log$ is the natural logarithm. Fix the parameters $γ = 2$ , $ε = 1 / 2$ , $p = 2 L / n$ , $d^{*} = 2 L$ , and $α = 8 / \sqrt{2 L}$ . For all sufficiently large such $n$ we have $0 < p < 1$ . Let $G \sim G (n, p)$ be the loop-free random graph on $[n]$ with independent Bernoulli- $p$ unordered-edge indicators; all probabilities below use this finite product law. Write $d (u) = \deg_{G} (u)$ , $d_{\min} = \min_{u \in [n]} d (u)$ , and $d_{\max} = \max_{u \in [n]} d (u)$ .

Let $c^{-} < 0 < c^{+}$ be the signed shifted roots, characterized by $2 h (c^{-} - 1 / 2) = 2 h (c^{+} + 1 / 2) = 3 / 2$ , where $h (x) = (1 + x) \log (1 + x) - x$ for $x > - 1$ . Define the degree event by $D_{n} = {(1 + c^{-} + α) p n \leq d_{\min} \leq d_{\max} \leq (1 + c^{+} - α) p n},$ and let $E_{n}$ be the event that $G$ is an $(n, d^{*}, α, c^{-}, c^{+})$ -expander, with the two-sided spectral expander predicate. For all sufficiently large $n$ , both success probabilities satisfy $P (D_{n}) \leq η_{n}, P (E_{n}) \leq η_{n}, η_{n} = 2 e^{4} L / \sqrt{n} + 2 p / (1 - p)^{2} ⟶ 0.$ Consequently, for neither event is there a constant $C > 0$ , independent of $n$ , such that its failure probability is at most $C (\log n)^{4} n^{- 1 / 2}$ for every sufficiently large $n$ .

Statement references

The Erdős-Rényi finite random graph G(n,p)

Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree

Shifted rate roots for quantitative expansion

Two-sided spectral expander — Definition 1.9

Facts & Assumptions

[F1]

For the graph $G \sim G (n, p)$ specified above, $P (d_{\min} > ⌊ L ⌋) \leq η_{n} \to 0$ for all sufficiently large $n$ (The gamma-two graph has a vertex of degree at most log n with high probability).

[F2]

For these admissible parameters, each of the full expander predicate and the degree target $D_{n}$ forces $d_{\min} > L$ for all sufficiently large $n$ (The original shifted roots impose a minimum degree above log n).

[F3]

For every $a > 0$ , $\log n / n^{a} \to 0$ (The logarithm grows more slowly than every positive real power).

[F4]

The graph law is the finite product of the Bernoulli- $p$ edge laws, and degree counts the incident edges in this loop-free graph; the degree extrema are defined on the nonempty set $[n]$ (The Erdős-Rényi finite random graph G(n,p), Adjacency, incidence, open and closed neighbourhoods, vertex degree, minimum degree and maximum degree).

[F5]

The shifted roots and the two-sided expander predicate used for $E_{n}$ are those in Shifted rate roots for quantitative expansion and Two-sided spectral expander — Definition 1.9, respectively, specialized to the parameters in the Statement.

[F6]

The floor satisfies $⌊ L ⌋ \leq L < ⌊ L ⌋ + 1$ . In a finite probability space, event containment implies the corresponding probability inequality, and $P (B^{c}) = 1 - P (B)$ (Integer part: for every real x there is exactly one integer m with m ≤ x < m + 1, Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space).

Proof

Given: The objects and hypotheses in the Statement.

Goal: Prove its conclusions.

1.1F1F2F3F4F5F6givenalgebra

By [F3], $p = 2 L / n \to 0$ , so for sufficiently large $n$ the graph law and degree extrema are defined as in [F4]. By [F5] the events $D_{n}, E_{n}$ are exactly those covered by [F2]. Hence $D_{n}, E_{n} \subseteq {d_{\min} > L} \subseteq {d_{\min} > ⌊ L ⌋}$ , using the floor inequality in [F6]. Applying probability monotonicity from [F6] and the bound [F1] gives both success-probability upper bounds. Their complement probabilities are at least $1 - η_{n}$ and at most one, so both failure probabilities tend to one.

2.1F1F3step 1.1algebra∎

For any fixed $C > 0$ , [F3] with $a = 1 / 8$ gives $C (\log n)^{4} / \sqrt{n} = C (\log n / n^{1 / 8})^{4} \to 0$ . Eventually $η_{n} < 1 / 4$ and this proposed failure bound is below $1 / 4$ . The actual failure probability of each event exceeds $3 / 4$ , contradicting any such bound beyond every sufficiently large cutoff. All parameters remain fixed except $n$ and its prescribed $p, d^{*}, α$ .

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.