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Kříž invokes the finite Ramsey theorem in the proofs of Theorems 3.3 and 4.1 and a compactness argument in the proof of Theorem 3.2. The following are the precise external forms used in this reconstruction. Their proofs are not included in this source unit.

Finite Ramsey theorem

For integers $q,k\ge1$ and $0\le r\le q$, there is an integer $m\ge q$ such that every map

$$\tau:\binom{[m]}r\longrightarrow[k]$$

has a set $M\subseteq[m]$ of size $q$ on which $\tau$ is constant on $\binom Mr$. For $r=0$ this is immediate. The nontrivial finite theorem is the classical Ramsey theorem cited as reference [3]: F. P. Ramsey, On a problem of formal logic, Proceedings of the London Mathematical Society 30 (1930), 264–286. Kříž's applications are on published pp. 904–905 (publisher PDF). The complete original finite proof is compiled at Ramsey (1930), Theorem B. It remains external to this Kříž source unit.

Theorem 3.3 uses $q=|G|$ and a single rank $r$. The generalized Theorem 3.4 uses successive applications for ranks $1,\ldots,|G|$; the finite iteration is explained there. Theorem 4.1 uses $q=t$, $r=t-1$, and $k^{n-1}$ colors, where $t$ is the length of a point orbit.

Finite-choice selection

Use the exact Rado selection principle quoted by de Bruijn–Erdős (1951), Theorem 2, printed p. 371. In the constant-palette form needed here, let $I$ be a set and let $K$ be a nonempty finite set. Suppose that for every finite $X\subseteq I$ a function $c_X:X\to K$ has been chosen. Then there is $c:I\to K$ such that for every finite $Y\subseteq I$, some finite $X\supseteq Y$ satisfies $c|_Y=c_X|_Y$.

The complete original proof is compiled at Rado (1949), Lemma 1, printed pp. 337–339. It remains external to this Kříž source unit.

The finite-witness lemma gives the complete application to equivalence Ramsey constraints. No graph-only compactness statement is silently substituted for those constraints. Choices of avoiding colorings and the stated selection principle are allowed.

What is proved locally

All same-paper product, orbit, and solvable-group deductions are expanded on their result pages. The finite group facts used to pass from a soluble group to a cyclic quotient are proved in Theorem 4.3. The geometric extension fact is proved in Observation 2.2.1.

The source cites Euclidean Ramsey Theorems I for the first three regular polyhedra. The proof of Corollary 4.6 instead supplies their elementary soluble symmetry groups and retains the source's distinct two-orbit argument for the remaining cases. No full proof of Paper I is claimed here.

Bears on. Problem 174.