Alphabeta Math

Ramsey Theory — 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.

✓ 8 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 8 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Combinatorics
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

R(3,3)=6 in both directions: the six-vertex argument and the red 5-cycle whose blue complement is another 5-cycle

Example

W(3,2)=9 by an explicit colouring of {0,…,7} and an exhaustive symmetry-reduced proof for {0,…,8}

Example

Infinite Ramsey for pairs gives a nondecreasing or nonincreasing subsequence of every real sequence

Example

Infinite Ramsey for triples gives a convex or concave subsequence of every real sequence in general position

False statement

FALSE: every two-colouring of ℕ contains an infinite monochromatic arithmetic progression

Counterexample

Finite strictly decreasing sequences of naturals form a tree with every finite level nonempty but no infinite branch

Counterexample

Infinite Ramsey fails with infinitely many colours: colour {i,j} by min{i,j}

Example

Constant, injective, left-dependent, and right-dependent pair colourings all occur on ℕ

Sources