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.
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- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
All subsets of of size at most : VC dimension and exactly members
Example
Let be the family of all subsets of of size at most :
Facts & Assumptions
Given: the family above.
Sauer-Shelah bounds a VC-dimension- family by (Sauer–Shelah: a family on of VC dimension at most has at most members, The set of -element subsets and the binomial coefficient ).
Verification
The set is shattered: every one of its subsets appears as a trace of the displayed family.
No three-element subset is shattered, because the three-element set itself is missing from the family and therefore cannot appear as a trace on that triple. Hence .
The family has exactly members, matching the bound of [L1].
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
- Shattering and the Vapnik–Chervonenkis dimension of a set family
- The down-shift $S_j$ of a set family at a point $j$
- $\lvert S_j(\mathcal{F})\rvert=\lvert\mathcal{F}\rvert$, and $w(S_j(\mathcal{F}))\le w(\mathcal{F})$ with equality only when $S_j(\mathcal{F})=\mathcal{F}$
- The set $[A]^{k}$ of $k$-element subsets and the binomial coefficient $\binom{n}{k} := \lvert [n]^{k}\rvert$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
19 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)