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 six -subsets of are -intersecting, and the bound holds
Example
The six -subsets of are
Facts & Assumptions
Given: the six pairs above.
An -intersecting family with has at most members (An -intersecting family on with has at most members).
Verification
Any two distinct displayed sets meet in either or point, so the family is -intersecting.
The family has six members, and [L1] gives the upper bound .
For example, the polynomial attached to over is It is nonzero at , where its value is , and vanishes at every other displayed pair, whose intersection with has size or .
Depends on
- An $L$-intersecting family on $[n]$ with $\lvert L\rvert=s$ has at most $\sum_{i=0}^{s}\binom{n}{i}$ members
- $L$-intersecting families
- Multilinear polynomials and the reduction $x_i^{2}\mapsto x_i$ on the cube
- $\widetilde f$ is multilinear, agrees with $f$ at every point of $\{0,1\}^{n}$, is degree-nonincreasing when nonzero, and is the unique multilinear polynomial with that agreement
- 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
26 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, §4.3 (standard reference, not scraped)