TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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The Ramsey number $R(3,3)=6$

Statement

The Ramsey number of The off-diagonal Ramsey number $R(s,t)$ as the least $N$ with $N\to(s,t)^2$ , for positive $s,t$ satisfies $R(3,3)=6$ . Complete graphs and cycles use Empty and complete graphs, complete bipartite graphs, and the convention that $P_n$ and $C_n$ have $n$ vertices and Walks, closed walks, trails, paths and cycles, with length equal to the number of traversed edges.

Facts & Assumptions

Given: Red-blue colourings of the edges of $K_6$ and $K_5$ .

[L1]

If $A,B$ are finite, $k\in\mathbb N$ , and $f:A\to B$ satisfies $\lvert A\rvert > k\,\lvert B\rvert$ , then there is $b \in B$ with $\lvert f^{-1}[\{b\}]\rvert > k$ (If $\lvert A\rvert > k\lvert B\rvert$ then every $f : A \to B$ has a fibre with more than $k$ elements, and for nonempty $B$ some fibre has at least $\lceil \lvert A\rvert / \lvert B\rvert\rceil$ elements).

Proof

technique · direct

1.1

At a fixed vertex $v$ of $K_6$ , at least three of its five incident edges have one colour by [L1]. Call their other endpoints $a,b,c$ and suppose that colour is red. If one of $ab,bc,ca$ is red it closes a red triangle with $v$ ; if none is red, then $a,b,c$ form a blue triangle. The same argument with the colour names exchanged proves $6\to(3,3)^2$ .

1.2

On five cyclically ordered vertices, colour the five cycle edges red and the remaining five edges blue. The red graph is a $5$ -cycle and has no triangle; the blue graph is also a $5$ -cycle, in the order obtained by stepping two places at a time, and has no triangle. Thus $5\not\to(3,3)^2$ .

construct

2.1

Step 1.1 gives $R(3,3)\le6$ and step 1.2 gives $R(3,3)>5$ . Since $R(3,3)$ is a natural number, it equals $6$ .

step 1.1step 1.2∎

Depends on

Used by

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

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 59 results over 18 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

Douglas West, Combinatorial Game Theory, Ramsey example (standard reference, not scraped)

The Ramsey number R(3,3)=6R(3,3)=6R(3,3)=6