CorollaryStatement: AI-generatedProof: AI-generatedprecheck passaudited 2026-08-11

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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.

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Infinite Ramsey holds for every set equipped with an injection from $N$

Statement

Let $X$ be a set equipped with an injection $j : N \to X$ . For every positive $k$ , every finite colouring of $[X]^{k}$ has an infinite monochromatic subset contained in $j [N]$ . The terms injection, equinumerous and monochromatic are those of Injection, surjection, bijection, Equinumerous sets, $A \approx B$ and $A ⪯ B$ and Finite colourings of $k$ -element subsets, monochromatic sets, and the arrow notations $N \to (s, t)^{2}$ and $N \to (r)_{c}^{k}$ .

Facts & Assumptions

Given: An injection $j : N \to X$ and a finite colouring $c : [X]^{k} \to C$ .

[L1]

Every finite colouring of $[N]^{k}$ has an infinite monochromatic set, in ZF (Infinite Ramsey theorem on $N$ : every finite colouring of $[N]^{k}$ has an infinite monochromatic set, in ZF).

[F1]

$f$ is injective (one-to-one) if $f (x) = f (y)$ implies $x = y$ (Injection, surjection, bijection).

Proof

technique · direct

1.1

Define a colouring of $[N]^{k}$ by $A \mapsto c (j [A])$ . Injectivity in [F1] makes $j [A]$ a $k$ -element set, so [L1] gives an infinite homogeneous $H \subseteq N$ .

L1F1

2.1

By [F1], the restriction $j ∣_{H}$ is a bijection from $H$ to $j [H]$ , so $j [H]$ is infinite. The pullback definition shows every $k$ -subset of $j [H]$ has the same $c$ -colour.

step 1.1F1∎

Depends on

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Infinite Ramsey holds for every set equipped with an injection from NN