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: distinct nonempty whose pairwise intersections all have the same parity satisfy
Statement
False claim: if distinct nonempty subsets of have pairwise intersections all of the same parity, then there are at most of them.
Facts & Assumptions
Given: the seven nonempty unions of the three pairs , and .
An Eventown family consists of distinct sets whose own sizes and pairwise intersection sizes are even (Eventown: distinct with every and every even satisfy ).
Refutation
Each chosen set and each pairwise intersection is a union of some of the three disjoint pairs, hence has even size. Thus the family satisfies the Eventown conditions of [L1], and all pairwise intersections have the same parity.
The family has distinct nonempty members on points. Since , it satisfies every hypothesis of the false claim and violates its conclusion.
Remarks
- The broken step is the real-positivity argument in Fisher's proof. Over there is no ordered notion of sum of squares.
Depends on
- Fisher's inequality, nonuniform form: distinct nonempty $A_1,\dots,A_m\subseteq[n]$ with $\lvert A_i\cap A_j\rvert=t$ for all $i\ne j$ satisfy $m\le n$
- If $v_1,\dots,v_m\in\mathbb{R}^{n}$ satisfy $\langle v_i,v_j\rangle=t\ge0$ for $i\ne j$ and $\langle v_i,v_i\rangle>t$, they are linearly independent
- Eventown: distinct $A_1,\dots,A_m\subseteq[n]$ with every $\lvert A_i\rvert$ and every $\lvert A_i\cap A_j\rvert$ even satisfy $m\le 2^{\lfloor n/2\rfloor}$
- The standard bilinear form $\langle x,y\rangle=\sum_{i<n}x_iy_i$ on $F^{n}$
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, §§2.3.2, 4.1 (standard reference, not scraped)