Proof By Lemma 2.2, there exists some $\theta (y)\in \operatorname {tp}(b/M)$ such that the elements of $\varphi (U,b)$ realize only finitely many $\varphi $ -types over $\theta (M)$ , and furthermore for any such type condition (a) or condition (b) in the lemma holds. By passing from $\theta (M)$ to $\theta (M) \cap \{ a \in M^{|y|} : \varphi (U,a)\subseteq \chi (U)\}$ if necessary, we may also assume that every $a\in \theta (M)$ satisfies that $\varphi (U,a)\subseteq \chi (U)$ . In particular, to prove Proposition 3.1 it suffices to find some $a\in \theta (M)$ such that $\varphi (U,a)\cap \varphi (U,b)=\emptyset $ . Since otherwise the result is trivial we may assume that $\varphi (U,b)\neq \emptyset $ .

Let $p_1(x), \ldots , p_l(x)$ denote the distinct $\varphi $ -types over $\theta (M)$ realized by elements of $\varphi (U,b)$ . We prove Proposition 3.1 by finding some $a\in \theta (M)$ such that $\varphi (x,a)\notin p_i(x)$ for every $i\leq l$ . If $l=1$ and $p_1(x)$ is the (unique) type described by condition (a) in Lemma 2.2, then clearly it suffices to take any $a\in \theta (M)$ and we are done. We assume this is not the case.

Let the numbering of the types $p_i(x)$ be such that, for some fixed $k \in \{l-1, l\}$ , the types $p_i(x)$ for $1\leq i \leq k$ satisfy condition (b) and the possibly remaining type $p_{i}(x)$ for $k<i\leq l$ satisfies condition (a) in Lemma 2.2. Hence, either $k=l$ or otherwise $1\leq k=l-1$ and the type $p_l(x)$ satisfies that $\varphi (x,a)\notin p_l(x)$ for every $a\in \theta (M)$ . In either case it suffices to find some $a\in \theta (M)$ with $\varphi (x,a)\notin p_i(x)$ for every $1\leq i \leq k$ .

Now let us fix, for every $1 \leq i \leq k$ , an $L(M)$ -formula $\chi _i(x)$ satisfying the following conditions:

• • $p_i(x)\models \chi _i(x)$ for every $i < k$ .

• • $p_j(x) \models \chi _{k}(x)$ for all $k \leq j \leq l$ .

• • $\chi _i(U)\cap \chi _j(U)=\emptyset $ for every $i<j\leq k$ .

We define, for any $1\leq m \leq k$ and elements $a_1,\ldots , a_{m-1} \in M^{|y|}$ , a set $\psi _{m}(M, a_1,\ldots , a_{m-1}) \subseteq \theta (M)$ as follows.

For $m=k$ , let $\psi _k(M, a_1,\ldots , a_{k-1})$ denote the set of all $a\in \theta (M)$ such that

$$\begin{align*}\varphi(U,a)\subseteq \bigcup_{i=1}^{k-1} (\varphi(U, a_i)\cap \chi_i(U))\cup \chi_k(U). \end{align*}$$

For $m < k$ , let $\psi _{m}(M, a_1,\ldots , a_{m-1})$ denote the set of all $a\in \theta (M)$ such that

$$\begin{align*}\varphi(U,a)\subseteq \bigcup_{i=1}^{m-1} (\varphi(U,a_i)\cap \chi_i(U)) \cup \bigcup_{i=m}^k \chi_i(U) \end{align*}$$

and moreover there exists two elements $a', a"\in \psi _{m+1}(M, a_1,\ldots , a_{m-1}, a)$ , with

$$\begin{align*}\varphi(U,a')\cap \varphi(U,a")\cap \chi_{m+1}(U)=\emptyset. \end{align*}$$

We now prove two claims regarding the set $\psi _{1}(M)$ that will yield Proposition 3.1, by showing the existence of some $a\in \theta (M)$ with $\varphi (x,a)\notin p_i(x)$ for every $i\leq k$ .

Claim 3.3 There exist $a, a'\in \psi _1(M)$ such that

$$\begin{align*}\varphi(U,a)\cap \varphi(U,a')\cap \chi_{1}(U)=\emptyset. \end{align*}$$

Proof

For any $m \leq k$ consider the following two statements $(\text {I}_m)$ and $(\text {II}_m)$ :

1. (I _m) Let $a_i \in M^{|y|}$ be such that $\varphi (x,a_i) \in p_i(x)$ , for $i < m$ , and let $a\in \theta (M)$ . Suppose that

   $$\begin{align*}\varphi(U,a)\subseteq \bigcup_{i=1}^{m-1} (\varphi(U, a_i)\cap \chi_i(U)) \cup \bigcup_{i=m}^k \chi_i(U) \end{align*}$$
   and
   $$\begin{align*}\varphi(x,a)\in p_m(x). \end{align*}$$
   Then
   $$\begin{align*}a\in \psi_m(M, a_1,\ldots, a_{m-1}). \end{align*}$$

2. (II _m) Let $a_i \in M^{|y|}$ be such that $\varphi (x,a_i) \in p_i(x)$ , for $i < m$ . Then there exist

   $$\begin{align*}a, a'\in \psi_m(M, a_1,\ldots, a_{m-1}) \end{align*}$$
   such that
   $$\begin{align*}\varphi(U,a)\cap \varphi(U,a')\cap \chi_{m}(U)=\emptyset. \end{align*}$$

We prove $(\text {I}_m)$ and $(\text {II}_m)$ for every $m\leq k$ using a reverse induction on m. Claim 3.3 is then given by $(\text {II}_1)$ .

Trivially $(\text {I}_k)$ holds by definition of $\psi _k(M, a_1,\ldots , a_{k-1})$ , even without the condition $\varphi (x,a)\in p_k(x)$ . We prove the remaining statements as follows. For $m\leq k$ , we derive $(\text {II}_m)$ from $(\text {I}_m)$ using Claim 3.2. For $m<k$ , we derive $(\text {I}_m)$ from $(\text {II}_{m+1})$ .

Proof of $(\text {I}_m)\Rightarrow (\text {II}_{m})$ for $m\leq k$ .

Let $\varphi (x,a_i)\in p_i(x)$ for $i < m$ . Let $\theta '(M)$ be the set of all $a\in \theta (M)$ such that

$$\begin{align*}\varphi(U,a)\subseteq \bigcup_{i=1}^{m-1} (\varphi(U,a_i)\cap \chi_i(U)) \cup \bigcup_{i=m}^k \chi_i(U). \end{align*}$$

Note that $\theta '(y)\in \operatorname {tp}(b/M)$ . By definition of $p_m(x)$ (see condition (b) in Lemma 2.2), the set A of all $a\in \theta '(M)$ with $ \varphi (x,a)\in p_m(x) $ is not definable (in M). By $(\text {I}_m)$ note that

$$\begin{align*}A \subseteq \psi_m(M, a_1,\ldots, a_{m-1}). \end{align*}$$

By Claim 3.2, the set $\psi _m(M, a_1,\ldots , a_{m-1})$ is definable. Since the subset A is not definable, there must exist some $a\in \psi _m(M, a_1,\ldots , a_{m-1})$ that is not in A, in particular

$$\begin{align*}\varphi(x,a)\notin p_m(x). \end{align*}$$

Now, by Lemma 2.3, there exists some $a'\in \theta (M)$ with

$$\begin{align*}\varphi(U,a')\subseteq \bigcup_{i=1}^{m-1} (\varphi(U,a_i) \cap \chi_i(U)) \cup (\chi_m(U)\setminus \varphi(U,a)) \cup \bigcup_{i=m+1}^k \chi_i(U) \end{align*}$$

such that

$$\begin{align*}\varphi(x,a')\in p_m(x). \end{align*}$$

(In the case $m=k=l-1$ Lemma 2.3 can still be applied because $\varphi (x,a)\notin p_l(x)$ by definition of the type $p_l(x)$ .) Once again by $(\text {I}_m)$ it follows that

$$\begin{align*}a'\in \psi_m(M, a_1,\ldots, a_{m-1}). \end{align*}$$

Finally, by construction note that

$$\begin{align*}\varphi(U,a)\cap \varphi(U,a') \cap \chi_m(U) = \emptyset. \end{align*}$$

Proof of $(\text {II}_{m+1})\Rightarrow (\text {I}_{m})$ for $m<k$ .

Let $\varphi (x,a_i)\in p_i(x)$ for $i < m$ , and $a\in \theta (M)$ be as described in $(\text {I}_m)$ . In particular we have that $\varphi (x,a)\in p_m(x)$ .

By $(\text {II}_{m+1})$ , there exist $ a', a"\in \psi _{m+1}(M, a_1,\ldots , a_{m-1}, a) $ such that

$$\begin{align*}\varphi(U,a')\cap \varphi(U,a")\cap \chi_{m+1}(U)=\emptyset. \end{align*}$$

But then by definition this means that $a\in \psi _m(M, a_1,\ldots , a_{m-1})$ .

Claim 3.4 Suppose that there exists some $a'\in \psi _1(M)$ with

$$\begin{align*}\varphi(x,a')\notin p_1(x). \end{align*}$$

Then there exists some $a\in \theta (M)$ satisfying that

$$\begin{align*}\varphi(x,a)\notin p_i(x) \text{ for every } 1 \leq i \leq k. \end{align*}$$

Proof

For any $m \leq k$ consider the following statement $(\text {B}_m)$ :

1. (B _m) Let $a_i \in M^{|y|}$ , $i<m$ , be such that there exist $ a'\in \psi _m(M, a_1,\ldots , a_{m-1}), $ with

   $$\begin{align*}\varphi(x,a')\notin p_m(x). \end{align*}$$

   Then there exists some $a\in \theta (M)$ with

   $$\begin{align*}\varphi(U,a)\subseteq \bigcup_{i=1}^{m-1} \left(\varphi(U,a_i)\cap \chi_i(U)\right) \cup \bigcup_{i=m}^k \chi_i(U) \end{align*}$$
   satisfying that
   $$\begin{align*}\varphi(x,a)\notin p_j(x) \text{ for every } m \leq j \leq k. \end{align*}$$

We prove $(\text {B}_{m})$ for every $m \leq k$ by reverse induction on m. Claim 3.4 then immediately follows from $(\text {B}_1)$ . Let $a_i$ , for $i<m$ , and $a'$ be as in $(\text {B}_{m})$ .

For the base case $m=k$ , it clearly suffices to take $a=a'$ . We assume that $m<k$ and show that $(\text {B}_{m+1}) \Rightarrow (\text {B}_{m})$ .

By definition of $\psi _m(M, a_1,\ldots , a_{m-1})$ , there exist $a", a"' \in \psi _{m+1}(M, a_1,\ldots , a_{m-1}, a')$ with

$$\begin{align*}\varphi(U,a") \cap \varphi(U,a"') \cap \chi_{m+1}(U)=\emptyset. \end{align*}$$

Without loss of generality we may assume that $\varphi (x,a")\notin p_{m+1}(x)$ . By $(\text {B}_{m+1})$ , we derive that there exists some $a\in \theta (M)$ such that

(3) $$ \begin{align} \varphi(U,a)\subseteq \bigcup_{i=1}^{m-1} \left(\varphi(U,a_i)\cap \chi_i(U)\right) \cup (\varphi(U,a')\cap \chi_m(U)) \cup \bigcup_{i=m+1}^k \chi_i(U) \end{align} $$

and

$$\begin{align*}\varphi(x,a)\notin p_j(x) \text{ for every } m < j \leq k. \end{align*}$$

However, since $\varphi (x,a')\notin p_m(x)$ , then by (3) it must also be that $\varphi (x,a)\notin p_m(x)$ .

We now complete the proof of the proposition. By Claim 3.3, let $a', a"\in \psi _1(M)$ be two elements such that $\varphi (U,a') \cap \varphi (U,a") \cap \chi _1(U)=\emptyset $ . Without loss of generality we may assume that $a'$ is such that $\varphi (x,a')\notin p_1(x)$ . By Claim 3.4 we conclude that there exists some $a\in \theta (M)$ satisfying that $\varphi (x,a)\notin p_i$ for every $i\leq k$ , as desired.

Proof of Theorem C

Let $\varphi (x,y)$ be an $L(M)$ -formula with $\pi ^*_\varphi (n)\in o(n^2)$ . We assume that $\varphi (x,y)$ does not partition into finitely many consistent families and derive that it does not have the $(\omega ,2)$ -property, i.e., we build a sequence $(a_n : 1\leq n<\omega )$ in $M^{|y|}$ such that the family $\{ \varphi (x,a_n) : 1 \leq n<\omega \}$ is pairwise inconsistent.

Hence we assume that $\varphi (x,y)$ satisfies that, for any finite collection of $L(M)$ -formulas $\{ \sigma _i(y) : 1 \leq i\leq m\}$ , if the family $\{\varphi (x,a) : a\in \sigma _i(M)\}$ is consistent for every $i\leq m$ , then there exists some $a\in M^{|y|}$ such that $a\notin \cup _i \sigma _i(M)$ . By model theoretic compactness we may fix some $b\in U^{|y|}$ satisfying that, for any formula $\sigma (y)\in \operatorname {tp}(b/M)$ , the family $\{\varphi (x,a) : a\in \sigma (M)\}$ fails to be consistent. We build our sequence $(a_n : 1 \leq n<\omega )$ using Proposition 3.1. In particular it will satisfy that, for every $i<\omega $ , it holds that

(4) $$ \begin{align} \varphi(U,a_i)\cap \varphi(U,b)=\emptyset. \end{align} $$

We proceed inductively on n.

By Proposition 3.1 (with $\chi (x):="x=x"$ ), let $a_1\in M^{|y|}$ be any element satisfying (4). Then, for the inductive step, let $(a_1,\ldots , a_{n-1})$ be elements each satisfying (4) and such that the formulas $\varphi (x,a_i)$ , for $i < n$ , are pairwise inconsistent. Let $\chi (x)$ denote the formula

$$\begin{align*}\bigwedge_{i=1}^{n-1} \neg\varphi(x,a_i). \end{align*}$$

Note that $\varphi (U,b)\subseteq \chi (U)$ . Now, applying Proposition 3.1, let $a_n\in M^{|y|}$ be an element satisfying (4) and $\varphi (U,a_n)\subseteq \chi (U)$ . The family $\{ \varphi (x,a_i) : 1 \leq i\leq n\}$ is pairwise inconsistent as desired.

1. (1) Definable $(\omega ,q)$ -conjecture: Let $\varphi (x,y)$ be a formula and let $q \geq 2$ be an integer such that $\pi ^*_\varphi (n)\in o(n^q)$ . If $\varphi (x,y)$ has the $(\omega ,q)$ -property, does it partition into finitely many consistent definable subfamilies?

2. (2) Uniform definable $(p,2)$ -conjecture 1: Let $\varphi (x,y)$ and $\psi (y,z)$ be formulas where $\pi ^*_\varphi (n)\in o(n^2)$ . Given any integer $p\geq 2$ , is there an m such that any family of the form $\{ \varphi (x,a) : M\models \psi (a,b) \}$ , for $b\in M^{|z|}$ , with the $(p,2)$ -property partitions into at most m consistent definable subfamilies?

3. (3) Uniform definable $(p,2)$ -conjecture 2: Let $\varphi (x,y)$ be a formula with $\pi ^*_\varphi (n)\in o(n^2)$ . Given any integer $p\geq 2$ , is there an m such that any definable subfamily of $\varphi (x,y)$ with the $(p,2)$ -property partitions into at most m consistent definable subfamilies?