DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

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.

The off-diagonal Ramsey number $R (s, t)$ as the least $N$ with $N \to (s, t)^{2}$ , for positive $s, t$

Definition

For positive natural numbers $s, t$ , the off-diagonal Ramsey number is

$R (s, t) : = \min {N \geq 1 : N \to (s, t)^{2}},$

where the arrow is defined in Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N \to (s, t)^{2}$ and $N \to (r)_{c}^{k}$ . The defining set is nonempty because Finite graph Ramsey theorem: $(\binom{s + t - 2}{s - 1}) \to (s, t)^{2}$ for all positive $s, t$ supplies the member $(\binom{s + t - 2}{s - 1})$ , and it has a least element by The well-ordering principle. Thus the notation presupposes neither an unproved existence claim nor a choice.

Depends on

Used by

R(s,t)≤ R(s-1,t)+R(s,t-1) for s,t≥2, and R(k,k)leC(2k-2, k-1)≤2²ᵏ⁻² Corollary
Turán graphs give the Ramsey lower bound R(s,t)≥(s-1)(t-1)+1 Corollary
The random-colouring proof of R(k,k)>2^k/2 Example
Erdős's finite counting bound R(k,k)>2^k/2 for every k≥3 Theorem
The Ramsey number R(3,3)=6 Theorem

Dependency tree · two levels

16 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

J. Fox et al., Graph Ramsey Theory, Section 2.1 (standard reference, not scraped)
R. Diestel, Graph Theory, 6th ed., Chapter 9, Section 9.1 (standard reference, not scraped)

The off-diagonal Ramsey number R(s,t)R(s,t) as the least NN with N→(s,t)2N→(s,t)2, for positive s,ts,t