---

Exact external inputs

Axiom of selections

The printed statement of Theorem A explicitly assumes the “axiom of selections.” Its proof uses that assumption to continue a countable sequence of choices from nested infinite sets.

The constructive direction of the serial-form theorem also invokes it when ordering an arbitrary infinite universe. All finite arguments are independent of this assumption. This compilation retains Ramsey's wording and does not identify it with a sharper modern choice principle.

Finite propositional normalization

The logic pages use the elementary facts that a truth function of finitely many propositions has a complete disjunctive normal form and that finitely many truth assignments can be enumerated. Ramsey cites Hilbert–Ackermann and mentions Wittgenstein in this connection. These are exact finite propositional inputs.

Elementary background

The proofs also use finite induction, the pigeonhole principle, permutation counting, the equality axioms, total ordering of a finite set, and the handshaking identity for a finite graph.

Results proved inside the paper

The infinite Ramsey theorem, the asymmetric finite two-color lemma, and the finite multicolor Ramsey theorem are all proved in the paper. In particular, the finite theorem used in the logical argument is not imported from the infinite theorem and is not obtained through compactness.

The truth-alternative reduction, the small-universe criterion, repeated-argument normalization, the serial-form theorem, the binary specialization, and the existential-before-universal reduction are likewise same-paper proofs reconstructed in this source unit.

Historical citations

Ramsey cites Behmann's unary-language decision result, Bernays–Schönfinkel's two-apparent-variable result without identity, and Langford's work on general-law postulate systems and an order special case. Those results explain the paper's setting but are not premises needed for the ten reconstructed components. They have not been recursively proof-reviewed here.