Ranks based on strong amalgamation Fraïssé classes

Abstract

In this paper, we introduce the notion of ${\textbf{K}}$-rank, where ${\textbf{K}}$ is a strong amalgamation Fraïssé class. Roughly speaking, the ${\textbf{K}}$-rank of a partial type is the number “copies” of ${\textbf{K}}$ that can be “independently coded” inside of the type. We study ${\textbf{K}}$-rank for specific examples of ${\textbf{K}}$, including linear orders, equivalence relations, and graphs. We discuss the relationship of ${\textbf{K}}$-rank to other ranks in model theory, including dp-rank and op-dimension (a notion coined by the first author and C. D. Hill in previous work).

Appendix A. Combinatorial lemmas

Fix $k < \omega$ and let

$$\begin{aligned} {\mathcal {D}} _k = \{ t \in {}^k \{ -1, 0, 1 \} : t(i) = 1 \text { for } i \text { minimal such that } t(i) \ne 0 \}. \end{aligned}$$

For ${\overline{a}} , {\overline{b}} \in \omega ^k$ and $t \in {\mathcal {D}} _k$, define

$$\begin{aligned} {\overline{a}} \le _t {\overline{b}} \text { if, for all } i< k, {\left\{ \begin{array}{ll} a_i < b_i &{} \text { if } t(i) = 1, \\ a_i = b_i &{} \text { if } t(i) = 0, \\ a_i > b_i &{} \text { if } t(i) = -1. \end{array}\right. } \end{aligned}$$

Finally, for all ${\overline{a}} , {\overline{b}} \in \omega ^k$, define

$$\begin{aligned} {\overline{a}} \le _\textrm{lex} {\overline{b}} \text { if } a_i < b_i \text { for } i \text { minimal such that } a_i \ne b_i. \end{aligned}$$

Note that ${\overline{a}} \le _\textrm{lex} {\overline{b}}$ if and only if there exists $t \in {\mathcal {D}} _k$ such that ${\overline{a}} \le _t {\overline{b}}$.

Lemma A.1

For all $k, \ell , m < \omega$, there exists $n < \omega$ such that, for all colorings $c : \left( {\begin{array}{c}n^k\\ \le 2\end{array}}\right) \rightarrow \ell$, there exist $Y_0, \dots , Y_{k-1} \in \left( {\begin{array}{c}n\\ m\end{array}}\right)$ such that, for all $t \in {\mathcal {D}} _k$, c is constant on the set

$$\begin{aligned} X_t = \left\{ \{ {\overline{a}} , {\overline{b}} \} : {\overline{a}} , {\overline{b}} \in \prod _{i < k} Y_i, {\overline{a}} \le _t {\overline{b}} \right\} . \end{aligned}$$

Proof

By induction on k. Let $k = 1$ and fix $\ell , m < \omega$. By Ramsey’s Theorem, there exists n such that, for all colorings $c : \left( {\begin{array}{c}n\\ \le 2\end{array}}\right) \rightarrow \ell$, there exists $Y \in \left( {\begin{array}{c}n\\ m\end{array}}\right)$ such that c is constant on $\left( {\begin{array}{c}Y\\ 1\end{array}}\right)$ and c is constant on $\left( {\begin{array}{c}Y\\ 2\end{array}}\right)$. Since $X_0 = \left( {\begin{array}{c}Y\\ 1\end{array}}\right)$ and $X_{1} = \left( {\begin{array}{c}Y\\ 2\end{array}}\right)$, this is the desired conclusion.

Fix $k, m, \ell < \omega$. Let

$$\begin{aligned} \ell ' = {}^{{\mathcal {D}} _k \times \{ -1, 1 \}} \ell . \end{aligned}$$

By Ramsey’s Theorem, there exists $n' < \omega$ such that, for all colorings $c' : \left( {\begin{array}{c}n'\\ \le 2\end{array}}\right) \rightarrow \ell '$, there exists $Y_k \in \left( {\begin{array}{c}n'\\ m\end{array}}\right)$ such that $c'$ is constant on $\left( {\begin{array}{c}Y_k\\ 1\end{array}}\right)$ and $c'$ is constant on $\left( {\begin{array}{c}Y_k\\ 2\end{array}}\right)$. Let

$$\begin{aligned} \ell '' = {}^{(n')^2} \ell . \end{aligned}$$

By the inductive hypothesis, there exists $n'' < \omega$ such that, for all colorings $c'' : \left( {\begin{array}{c}(n'')^k\\ \le 2\end{array}}\right) \rightarrow \ell ''$, there exist $Y_0, \dots , Y_{k-1} \in \left( {\begin{array}{c}n''\\ m\end{array}}\right)$ such that, for all $t \in {\mathcal {D}} _k$, $c''$ is constant on $X_t$. Let $n = \max \{ n', n'' \}$.

Fix a coloring $c : \left( {\begin{array}{c}n^{k+1}\\ \le 2\end{array}}\right) \rightarrow \ell$. This induces a coloring $c'' : \left( {\begin{array}{c}(n'')^k\\ \le 2\end{array}}\right) \rightarrow \ell ''$ given by: for each ${\overline{a}} , {\overline{b}} \in (n'')^k$ with ${\overline{a}} \le _\textrm{lex} {\overline{b}}$, for each $i, j \in n'$, let

$$\begin{aligned} c''( \{ {\overline{a}} , {\overline{b}} \} )(i, j) = c( \{ {\overline{a}} {}^{\frown } i, {\overline{b}} {}^{\frown } j \} ). \end{aligned}$$

Thus, there exist $Y_0, \dots , Y_{k-1} \in \left( {\begin{array}{c}n''\\ m\end{array}}\right)$ such that, for all $t \in {\mathcal {D}} _k$, $c''$ is constant on $X_t$. Now define $c' : \left( {\begin{array}{c}n'\\ \le 2\end{array}}\right) \rightarrow \ell '$ as follows: for each $i \le j < n'$, $t \in {\mathcal {D}} _k$, and $s \in \{ -1, 1 \}$, choose ${\overline{a}} , {\overline{b}} \in \prod _{i < k} Y_i$ with ${\overline{a}} \le _t {\overline{b}}$ and set

$$\begin{aligned} c'( \{ i, j \} )(t,s) = {\left\{ \begin{array}{ll} c( \{ {\overline{a}} {}^{\frown } i, {\overline{b}} {}^{\frown } j \} ) &{} \text { if } s = 1, \\ c( \{ {\overline{a}} {}^{\frown } j, {\overline{b}} {}^{\frown } i \} ) &{} \text { if } s = -1. \end{array}\right. } \end{aligned}$$

Since $c''$ is constant on $X_t$ for each t, this function is independent of the choice of ${\overline{a}}$ and ${\overline{b}}$. Thus, there exists $Y_k \in \left( {\begin{array}{c}n'\\ m\end{array}}\right)$ such that $c'$ is constant on $\left( {\begin{array}{c}Y_k\\ 1\end{array}}\right)$ and $c'$ is constant on $\left( {\begin{array}{c}Y_k\\ 2\end{array}}\right)$. We claim that $Y_0, \dots , Y_k$ work for c.

Fix $t \in {\mathcal {D}} _{k+1}$. If $t(k)=0$, let

$$\begin{aligned} r = c'( \{ i \} )(t |_k, 1 ) \end{aligned}$$

for any choice of $i \in Y_k$. Since $c'$ is constant on $\left( {\begin{array}{c}Y_k\\ 1\end{array}}\right)$, this is independent of the choice of i. If $t(k) \ne 0$, let

$$\begin{aligned} r = c'( \{ i, j \})(t |_k, t(k)) \end{aligned}$$

for any choice of $i, j \in Y_k$ with $i < j$. Since $c'$ is constant on $\left( {\begin{array}{c}Y_k\\ 2\end{array}}\right)$, this is independent of the choice of i and j. Then, for any ${\overline{a}} , {\overline{b}} \in \prod _{i \le k} Y_i$ such that ${\overline{a}} \le _t {\overline{b}}$, we have that

$$\begin{aligned} c( \{ {\overline{a}} , {\overline{b}} \} ) = r. \end{aligned}$$

This is what we wanted to prove. $\square$

Corollary A.2

For all $k, \ell , m < \omega$, there exists $n < \omega$ such that, for all colorings $c : n^k \rightarrow \ell$, there exist $Y_0, \dots , Y_{k-1} \in \left( {\begin{array}{c}n\\ m\end{array}}\right)$ such that c is constant on $\prod _{i < k} Y_i$.

Proof

Since any coloring $c : n^k \rightarrow \ell$ can be extended arbitrarily to a coloring $c : \left( {\begin{array}{c}n^k\\ \le 2\end{array}}\right) \rightarrow \ell$, this follows immediately from Lemma A.1.

$\square$