CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Turán graphs give the Ramsey lower bound $R (s, t) \geq (s - 1) (t - 1) + 1$

Statement

For integers $s, t \geq 2$ ,

$R (s, t) \geq (s - 1) (t - 1) + 1.$

Facts & Assumptions

Given: The hypotheses and notation of the statement above.

[F1]

$N \to (s, t)^{2}$ means every red-blue colouring of the pairs of an $N$ -element set has a red $s$ -set or a blue $t$ -set (Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N \to (s, t)^{2}$ and $N \to (r)_{c}^{k}$ ).

[F2]

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

Proof

technique · exhibit a colouring below the claimed threshold

1.1

Partition $(s - 1) (t - 1)$ vertices into $s - 1$ parts of size $t - 1$ . Colour every edge within a part blue and every edge between parts red. A red clique uses at most one vertex from each part, so has size at most $s - 1$ ; a blue clique lies in one part, so has size at most $t - 1$ .

givenF1

2.1

Thus this colouring has neither a red $K_{s}$ nor a blue $K_{t}$ . The Ramsey-number definition makes $R (s, t)$ strictly larger than $(s - 1) (t - 1)$ , proving the integer lower bound.

step 1.1givenF1F2∎

Depends on

Used by

A Turán-partition colouring witnesses R(4,3)≥7 Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 26 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Yufei Zhao, Graph Theory and Additive Combinatorics (standard reference, not scraped)

Turán graphs give the Ramsey lower bound R(s,t)≥(s−1)(t−1)+1R(s,t)≥(s−1)(t−1)+1