ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11

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

Example

Every alternative in Canonical Ramsey theorem for pairs: on an infinite subset a colouring is constant, injective, left-dependent, or right-dependent is realised on the whole of $\mathbb N$ .

Facts & Assumptions

Given: Every unordered pair is written uniquely as $\{i,j\}$ with $i<j$ .

[L1]

On an infinite subset a pair-colouring is constant, injective, left-dependent, or right-dependent (Canonical Ramsey theorem for pairs: on an infinite subset a colouring is constant, injective, left-dependent, or right-dependent).

Verification

technique · direct

1.1

The formula $c_0(\{i,j\})=0$ is constant. The formula $c_I(\{i,j\})=\{i,j\}$ is injective because equal two-element subsets are the same unordered pair.

L1construct

2.1

For $i<j$ , set $c_L(\{i,j\})=i$ and $c_R(\{i,j\})=j$ . Then $c_L(\{i,j\})=c_L(\{k,l\})$ if and only if $i=k$ , and $c_R(\{i,j\})=c_R(\{k,l\})$ if and only if $j=l$ . Thus all four mutually distinct equality patterns listed in [L1] occur.

step 1.1L1∎

Depends on

Canonical Ramsey theorem for pairs: on an infinite subset a colouring is constant, injective, left-dependent, or right-dependent

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 9 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

I. B. Leader, Ramsey Theory, Theorem 4 and following remark (standard reference, not scraped)

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