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 induced-embedding count
Definition
For finite graphs and , define the induced-embedding count
The set inside the cardinality is a subset of the finite function set , so the displayed natural number is well defined (The set of functions between finite sets is finite, with , A subset of a finite set is finite, with , and equality holds if and only if , The cardinality of a finite set).
This convention counts labelled embeddings, not vertex subsets. An induced copy with image contributes one embedding for each isomorphism (Induced embeddings and induced copies of a graph).
Depends on
- Induced embeddings and induced copies of a graph
- The set $A^{B}$ of functions $B \to A$ between finite sets is finite, with $\lvert A^{B}\rvert = \lvert A\rvert^{\lvert B\rvert}$
- A subset of a finite set is finite, with $\lvert B\rvert \le \lvert A\rvert$, and equality holds if and only if $B = A$
- The cardinality $\lvert A\rvert$ of a finite set
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 54 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
- Swastik Kopparty, Local Structure: Subgraph Counts I (standard reference, not scraped)