ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-11

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

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 $N$ .

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 · two levels

5 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

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 NN