---

Source. These are elementary consequences of the definitions on Kříž's published p. 901, used implicitly in the conclusions of Theorems 3.3 and 4.1 (publisher PDF). This page expands those deductions; it is not a separately numbered source theorem.

Statement

Let $F$ be a finite configuration with equivalence relation $E$.

1. The equality relation is always Ramsey on $F$.
2. If $E'\subseteq E$ and $F$ is $E$-Ramsey, then $F$ is $E'$-Ramsey.
3. If $F$ is $E$-Ramsey, every subset $A\subseteq F$ is Ramsey for $E\cap(A\times A)$.
4. Congruent configurations have the same Ramsey properties, with their equivalence relations transported by the congruence.
5. If $\lambda>0$, then $F$ is $E$-Ramsey if and only if $\lambda F$ is $\lambda E$-Ramsey, where $\lambda E=\{(\lambda x,\lambda y):xEy\}$.

In particular, a subset of a Ramsey configuration is Ramsey. A configuration that is $E$-Ramsey with at most $s$ equivalence classes is $s$-Ramsey.

Full proof

For the equality relation, embed $F$ in any Euclidean space of sufficient dimension. The condition on equal points holds for every coloring. If $E'\subseteq E$, an embedding whose colors are constant on every $E$-class also satisfies all the $E'$-equalities. Restricting the same embedding to $A$ proves the subset assertion. Composing with a fixed congruence proves invariance under congruence.

For scaling, suppose first that $F$ is $E$-Ramsey and fix $k$. Choose a dimension $N$ that witnesses this property. Given $c:\mathbb R^N\to[k]$, apply the property of $F$ to $c_\lambda(v)=c(\lambda v)$. If $\phi:F\to\mathbb R^N$ is the resulting isometrical embedding, define

$$\psi(\lambda x)=\lambda\phi(x)\qquad(x\in F).$$

This is an isometrical embedding of $\lambda F$, because both domain and image distances are multiplied by $\lambda$. For $xEy$, its two colors are $c_\lambda(\phi(x))$ and $c_\lambda(\phi(y))$, which agree. Thus $\lambda F$ is $\lambda E$-Ramsey. Apply the same argument with $1/\lambda$ for the converse.

Finally, if each of at most $s$ classes is monochromatic, their union uses at most $s$ colors. No condition that distinct classes have distinct colors is needed. $\square$

Uses. Theorem 3.3, Theorem 4.1, and the subconfiguration consequence of Theorem 4.3.

Bears on. Problem 174.