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 shifting proof of Sauer–Shelah uses no field and no vector space
Remarks
The proof of Sauer–Shelah: a family on of VC dimension at most has at most members uses only the down-shifts, the weight function, the shattered-set definition and a count of subsets. No incidence vector, field, bilinear form or dimension appears there.
By contrast, A second proof of Sauer–Shelah, from the multilinear polynomial space does use the page's linear-algebra machinery: it works in a polynomial function space, compares spans of monomials, and closes by a dimension count. The page keeps both proofs because the bound belongs in this topic, but only one of the two routes spends the linear-algebra apparatus built here.
Depends on
- Sauer–Shelah: a family on $[n]$ of VC dimension at most $d$ has at most $\sum_{i=0}^{d}\binom{n}{i}$ members
- A second proof of Sauer–Shelah, from the multilinear polynomial space
- Every set shattered by $S_j(\mathcal{F})$ is shattered by $\mathcal{F}$
- Shattering and the Vapnik–Chervonenkis dimension of a set family
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
23 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
- L. Babai and P. Frankl, Linear Algebra Methods in Combinatorics, §7.4 (standard reference, not scraped)