ExampleConstruction: AI-generatedVerification: AI-generatedprecheck passaudited 2026-08-26

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.

A family containing $K_{1}$ is viral for vacuous reasons

Example

Every finite family of graphs containing $K_{1}$ is viral, but only vacuously: for $ϵ \in (0, \frac{1}{2})$ no nonempty graph satisfies the required $K_{1}$ -copy bound.

Facts & Assumptions

Given: A finite family $F$ with $K_{1} \in F$ , a real $ϵ \in (0, \frac{1}{2})$ , and a nonempty finite graph $G$ .

[L1]

Virality asks for the implication in The viral property for a finite forbidden family.

[L2]

${ind}_{K_{1}} (G)$ counts the induced embeddings of the one-vertex graph in $G$ (The induced-embedding count ${ind}_{H} (G)$ , Empty and complete graphs, complete bipartite graphs, and the convention that $P_{n}$ and $C_{n}$ have $n$ vertices).

[L3]

For a positive base, $ϵ^{d} = \exp (d \log ϵ)$ ; the logarithm and exponential are strictly increasing, and the exponential is positive (Real powers for positive bases, with the zero-base positive-exponent convention, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The exponential function is strictly increasing, The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ ).

Verification

technique · direct

1.1L2

Each vertex of $G$ determines one induced embedding of $K_{1}$ into $G$ , so [L2] gives ${ind}_{K_{1}} (G) = ∣ V (G) ∣$ .

2.1step 1.1L3algebra

If $d \geq 1$ , then $0 < ϵ < 1$ gives $\log ϵ < 0$ and therefore $d \log ϵ \leq \log ϵ < 0$ . By [L3], $0 < ϵ^{d} \leq ϵ < 1$ , so $ϵ^{d} ∣ V (G) ∣ < ∣ V (G) ∣ = {ind}_{K_{1}} (G)$ . Thus the defining viral inequality for $K_{1}$ can hold for no nonempty graph.

3.1step 2.1L1∎

Since the antecedent in [L1] has no nonempty instance, every exponent $d \geq 1$ witnesses the viral implication vacuously. Therefore any finite family containing $K_{1}$ is viral.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

26 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.

A family containing K1K1​ is viral for vacuous reasons