Alphabeta MathMatchings, Covers, Menger and Network Flows — Examples
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How 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.
✓ 6 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 6 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Real Analysis
2 · Logical flowchart
3 · Items
Counterexample
The middle edge of P₄ is a maximal matching but the two end edges form a larger matching
Example
Three left vertices with only two collective neighbours fail Hall's condition and cannot all be matched
Example
A three-set family with an explicit system of distinct representatives
Counterexample
K₃ has ν(K₃)=1<2=τ(K₃), so König's equality needs bipartiteness
Example
A four-vertex integral network with an exhibited maximum flow and minimum cut of value five
Example