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.
FALSE: a family on of VC dimension at most has at most members
Statement
False claim: a family on of VC dimension at most has at most members.
Facts & Assumptions
Given: the family on .
Sauer-Shelah gives the exact bound at , (Sauer–Shelah: a family on of VC dimension at most has at most members).
Refutation
The family has VC dimension : it shatters and nothing larger.
It has two members, while . So the false claim already fails at , .
Remarks
- The true polynomial estimate on the page is , not .
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
- For $d\ge1$ a family on $[n]$ of VC dimension at most $d$ has at most $(n+1)^{d}$ members
- Shattering and the Vapnik–Chervonenkis dimension of a set family
- The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$
- Exponentiation of natural numbers, $m^{n}$, and its agreement with the integer power in $\mathbb{R}$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
34 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)