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The canonical version is the published 1973 paper, pp. 341–363, read in the published scan named on the source card. The following are external inputs or background statements, not additional full proofs supplied by this source compilation.

Ramsey, arithmetic progressions and homothety

Theorem 1, p. 342. Given positive integers $k,\ell,r$ with $k\le\ell$, there is an $n$ such that every $r$-coloring of the $k$-subsets of an $n$-element set has an $\ell$-element subset all of whose $k$-subsets have one color. The source cites F. P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society (2) 30 (1930), 264–286. The complete original finite proof is compiled at Ramsey (1930), Theorem B. It remains external to this 1973 source unit. It is introductory background; the finite geometric deductions compiled here do not silently count its proof as a same-paper argument.

Theorem 2, p. 343. For every positive $L,r$ there is an $N$ such that every $r$-coloring of $\{1,\ldots,N\}$ has a monochromatic arithmetic progression of $L$ terms with positive integer common difference. This is the exact van der Waerden theorem used in theorem_17. The paper cites B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Archief voor Wiskunde 15 (1927), 212–216. The finite statement is explicit on p. 343; the original 1927 proof is not reconstructed here.

Theorem 3, p. 343. Every finite coloring of $\mathbb R^d$ contains a monochromatic homothetic copy $a+\lambda K$, $\lambda>0$, of any prescribed finite $K\subseteq\mathbb R^d$. This Gallai theorem allows the scale to vary, unlike the congruence questions. The source names it after Gallai and cites R. Rado, Note on combinatorial analysis, Proceedings of the London Mathematical Society (2) 48 (1943), 122–160, for it; it cites Hales–Jewett (1963) and Graham–Rothschild (1971) as other generalizations of van der Waerden's theorem. It is background, not an input needed to replace any omitted geometric step.

Compactness and ordinary algebra

The compactness input is that a product of finite discrete spaces is compact. Equivalently, any family of closed finite-coordinate coloring constraints with the finite intersection property has a simultaneous solution. Proposition 4 cites J. R. Shoenfield, Mathematical Logic (1967), p. 69. The exact deduction to finite forcing witnesses is fully given in compactness. The few-color extension uses the same explicit finite-coordinate argument.

The field proof uses the ordinary vector-space basis principle, including a basis containing a prescribed nonzero vector, and elementary finite linear algebra. Its rational, transcendental, finite-algebraic and arbitrary-field reductions are all proved in theorem_16. The field-coloring theorem is not an external black box in this compilation.

Seven-color negative result and reused finite geometry

Theorem 5 on p. 344 states that $R(P,2,7)$ is false for a pair at any fixed positive distance, whereas $R(P,2,3)$ is true. Its seven-color construction is referred to H. Hadwiger, H. Debrunner and V. Klee, Combinatorial Geometry in the Plane (1964), and L. Moser and W. Moser, Problem 10, Canadian Mathematical Bulletin 4 (1961), 187–189. The book and original problem solution have not been audited here. Thus the seven-color negative is an attributed external result, not a newly reconstructed proof.

The same seven-point spindle appears in Paper II. Its exact coordinates and independence bound already have one canonical complete proof at seven_point_spindle. The positive three-color deduction in theorem_5 uses that proof rather than duplicating the construction.

The p. 351 comment concerning other “nice” surfaces invokes L. O'Connor's 1972 UCLA thesis. No general surface-embedding theorem from that thesis is proved or needed for the numbered chains compiled here.

These citations identify what the 1973 source imports. Except for the already compiled Ramsey proof and spindle, they do not assert complete independent audits of the original referenced works.