Alphabeta MathExtremal Graph Theory: Examples and False Statements
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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.
✓ 9 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 9 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Combinatorics
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
2 · Logical flowchart
3 · Items
Example
T_10,3=K_3,3,4 has 33 edges and is the unique 10-vertex K₄-extremal graph
Example
K_3,4 realises ex(7,K₃)=12
Example
Deleting one edge from K_3,4 gives a triangle-free graph one edge below the Mantel threshold
Example
A Turán-partition colouring witnesses R(4,3)≥7
Example
The five-cycle is K_2,2-avoiding and shows the KST problem is not just a complete-bipartite construction
False statement
Every triangle-free graph is bipartite
Example
The Petersen graph has chromatic number 3, so its Turán density is 1/2
Example
Every odd cycle C₂ₖ₊₁ has Turán density 1/2
False statement