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: an Oddtown family on has at most members
Statement
False claim: every Oddtown family on has at most members.
Facts & Assumptions
Given: the singleton family on .
The singleton family is an Oddtown family of size exactly (The singletons form an Oddtown family, so the bound is attained for every ).
Refutation
By [L1], the family is an Oddtown family with members.
Since is not at most , the false claim fails for every .
Depends on
- The $n$ singletons form an Oddtown family, so the bound $m\le n$ is attained for every $n$
- Oddtown: distinct $A_1,\dots,A_m\subseteq[n]$ with every $\lvert A_i\rvert$ odd and every $\lvert A_i\cap A_j\rvert$ ($i\ne j$) even satisfy $m\le n$
- If the incidence vectors of $A_1,\dots,A_m\subseteq[n]$ are linearly independent over $F$ then $m\le n$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
17 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, §1.1 (standard reference, not scraped)