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The down-shift of a set family at a point
Definition
Let and let .
For a set , define
The down-shift of at is
Also define the weight
The next lemma shows that has the same number of members as and never larger weight.
Remarks
- The family is shifted only when the lower set is not already present. That is the clause that keeps the operation from collapsing two sets to one.
Depends on
Used by
- All subsets of [4] of size at most 2: VC dimension 2 and exactly ∑_i≤2C(4, i)=11 members Example
- | Sⱼ(F)|=lvertF|, and w(Sⱼ(F))≤ w(F) with equality only when Sⱼ(F)=F Lemma
- Applying down-shifts until none changes the family terminates, and the result is closed under taking subsets Lemma
- Every set shattered by Sⱼ(F) is shattered by F Lemma
- Sauer–Shelah: a family on [n] of VC dimension at most d has at most ∑ᵢ₌₀ᵈC(n, i) members Theorem
Dependency tree · two levels
24 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, §7.5 (standard reference, not scraped)