The computable strength of Milliken's Tree Theorem and applications

Paul-Elliot Anglès d'Auriac

Joint work with Peter Cholak, Damir Dzhafarov, Benoît Monin, Ludovic Patey

Third Workshop on Digitalization and Computable Models,

July 1, 2021

July 1, 2021

- The combinatorial relations between
- Devlin's theorem,
- The Rado graph theorem
- Milliken's tree theorem

- The computable strength of Milliken's Tree Theorem and two applications

Given a statement from ordinary mathematics, what is the optimal set of axioms to prove it?

Our "running" example from ordinary mathematics will be Ramsey's theorem:

Let \(f:[\mathbb N]^n\to k\) be a coloring of \(n\)-elements subsets, into \(k\) colors. Then, there exists an infinite \(S\subseteq \mathbb N\) such that \(f\) is monochromatic on \([S]^n\).

A set of axioms \(A\) is:

- An upper bound if the theorem can be proven with \(A\),
- A lower bound if the theorem proves \(A\).

Theorems in ordinary mathematics usually come in a \(\Pi^1_2\) form: \[\forall I,\exists S, P(I,S)\] which can be seen as a problem, with instances \(I\) and solutions \(S\).

- Instance: a coloring \(f:[\mathbb N]^n\to k\),
- Solution: an infinite set \(S\) homogeneous for \(f\)

In the computability-theoretic perspective of reverse mathematics, we are particularly interested in the complexity of the solutions relative to the instance.

- Upper bound: Every computable instance of this problem has a solution sufficiently easy to compute
- Lower bound: There is a computable instance of this problem with all solutions having high computational power

In order to prove non-implication of \(P\) to \(Q\), one usually needs to create a model of \(P\) that is not a model of \(Q\).

In the computability-theoretic perspective, the counterpart is:

- Every computable instance has a solution of sufficiently low computational power

A \(\Pi^1_2\) principle \(P\) admits cone avoidance if for every uncomputable \(A\), every computable instance of \(P\) admits a solution \(H\) which does not compute \(A\).

A \(\Pi^1_2\) principle \(P\) admits cone avoidance if for every \(Z\) and every \(A\not\leq_TZ\), every \(Z\)-computable instance of \(P\) admits a solution \(H\) with \(A\not\leq_TZ\oplus H\).

Ramsey's theorem for 2-tuples admits cone avoidance.

Ramsey's theorem for 2-tuples does not imply \(\mathrm{ACA}_0\).

Let \(f:[\mathbb N]^n\to k\) be a coloring of \(n\)-elements subsets, into \(k\) colors. Then, there exists an infinite \(S\subseteq \mathbb N\) such that \(f\) is monochromatic on \([S]^n\).

Adding structure to Ramsey's theorem would yield the following statement:

Let \(\mathbb A\) be a countable \(\mathcal L\)-structure on a language \(\mathcal L\), let \(f:[\mathbb A]^n\to k\) be a coloring of \(n\)-elements subsets of \(\mathbb A\).

Then, there exists a subcopy \(\mathbb S\subseteq \mathbb A\) such that \(f\) is monochromatic on \([\mathbb S]^n\).

Let \(\mathbb A\) be a countable \(\mathcal L\)-structure on a language \(\mathcal L\), let \(f:[\mathbb A]^n\to k\) be a coloring of \(n\)-elements subsets of \(\mathbb A\).

Then, there exists a subcopy \(\mathbb S\subseteq \mathbb A\) such that \(f\) takes at most \(\ell\) colors on \([\mathbb S]^n\).

- In the case of Ramsey's theorem, the language \(\mathcal L\) is empty. A subcopy of a countable \(\mathcal L\)-structure is an infinite subset.
- However, the statement is false for many structures, such as for 2-tuples when \(\mathbb A\) is a non-trivial graph, or whenever there are two non isomorphic substructures of cardinality \(n\). Thus, one needs to allow more colors in the output.

It is more natural for a Ramsey statement for structures to color \([\mathbb A]^{\mathbb F}\) the set of finite subcopies of \(\mathbb F\) inside \(\mathbb A\), where \(\mathbb F\) is a finite structure. We won't focus here on this distinction.

- An \(\mathcal L\)-structure \(\mathbb A\) has finite big Ramsey numbers if for every \(n\), there exists \(\ell_n\) such that Ramsey's statement for \(\mathbb A\), \(n\)-tuples and \(\ell_n\) colors allowed is true.
- The optimal \(\ell_n\) is called the big Ramsey number for \(n\)-tuples.

We will essentially see two structures with finite big Ramsey numbers:

A dense linear order is an \(\mathcal L\)-structure on the language \(\{\leq\}\), satisfying the axioms of a linear order and \(\forall x\lt y,\ \exists a,b,c,\) s.t. \[a\lt x\lt b\lt y\lt c\]

- An \(\mathcal L\)-structure \(\mathbb A\) is universal for a class of finite structures \(\mathcal C\) if every element of \(\mathcal C\) embeds into \(\mathbb A\).
- An \(\mathcal L\)-structure \(\mathbb A\) is homogeneous if every isomorphism between two finite substructures of \(\mathbb A\) extends to an automorphism of \(\mathbb A\).

A dense linear order is the homogeneous structure universal for the class of finite orders.

All \(n\)-tuples of elements of a DLO are isomorphic. However, the big Ramsey number for 2-tuples is not 1.

Consider a DLO \((Q,\leq_Q)\) with \(Q\subseteq\mathbb N\), and the 2-coloring \(f:[Q]^2\to 2\) such that: \[\forall x\lt_{\mathbb N}y\in Q,\quad f(\{x,y\})=0\quad\text{ iff }\quad x\leq_Q y\] Then\(f\) cannot be monochromatic on sub DLO \(Q'\) of \(Q\) as there would exists \(x,y_0, y_1\in Q'\) such that \[x\lt_{\mathbb N}y_0, y_1\quad\land\quad y_0\lt_Qx\lt_Qy_1\]

The DLO structure has finite big Ramsey numbers. The values are \(\ell_1=1\), \(\ell_2=2\), \(\ell_3=16\), \(\ell_4=272\), ...

A Rado graph \(\mathbb G\) is the \(\mathcal L\)-structure on \(\{E\}\) that satisfies the following:

For every disjoint finite \(F_0, F_1\subseteq\mathbb G\), there exists \(v\in\mathbb G\) such that: \[\forall v'\in F_0,\ v\mathbin Ev'\quad\quad\forall v'\in F_1,\ \lnot (v\mathbin Ev')\]

\(\mathbb G\) is homogeneous and universal for the class of finite graphs.

The big Ramsey number of the Rado graph for 2-tuples must be at least 2, as witnessed by the coloring which depends only on the substructure: \(f:[\mathbb G]^2\to 2\) s.t. \[f(\{x;y\})=0\iff x\mathbin E y\]

Any Rado subgraph of \(\mathbb G\) must contain two linked vertices and two non-linked vertices.

The Rado graph \(\mathbb G\) has finite big Ramsey numbers. The values are \(\ell_1=1\), \(\ell_2=4\), \(\ell_3=112\), \(\ell_4=12352\), ...

The proofs of both Devlin's theorem and the Rado graph theorem have the same structure. Their combinatorial core is the use of Milliken's tree theorem.

- A tree is a set \(T\subseteq \omega^{<\omega}\) closed under "longest common prefix".
- The level of a node \(\sigma\in T\) is the number of strict predecessors. The \(n\)-th level \(T(n)\) of a tree is the set of nodes of level \(n\).
- The height of a tree is the smallest ordinal strictly greater than the levels of all of its nodes
- A node \(\sigma\in T\) is \(k\)-branching if it has exactly \(k\) incompatible extensions in \(T\)

A tree \(S\subseteq T\) of height \(\alpha\leq\omega\) is a strong subtree of a tree \(T\) if:

- Levels are preserved: \(\exists f:\alpha\to\omega\) (the level function) such that \[\forall n<\alpha, \sigma\in S(n)\implies \sigma\in T(f(n))\]
- Branching is preserved: \(\forall\sigma\in S\) not in the last level of \(S\), \[\sigma\text{ is \(k\)-branching in \(S\)} \iff \text{\(\sigma\) is \(k\)-branching in \(T\)}\]

Let \(T\) be a tree, \(n\in\omega\) and \(f:\mathcal S_n(T)\to k\) a \(k\)-coloring of the strong subtrees of height \(n\).

Then, there exists \(S\in\mathcal S_\omega(T)\) a strong subtree of \(T\) of infinite height such that \(f\) is constant on \(\mathcal S_n(S)\).

We now glance at the common structure of the proof of Devlin's theorem and the Rado graph theorem, assuming Milliken's tree theorem.

Let \(\sigma,\tau\in2^{<\omega}\).
We say that \(\sigma\lt_{\mathbb Q}\tau\) if
\[(\sigma\wedge\tau)^\frown0\prec\sigma\qquad\text{or}\qquad (\sigma\wedge\tau)^\frown1\prec\tau\]

Proof of Devlin's Theorem: Let \(T\) be a perfect binary tree and \(S\) be any strong subtree of \(T\). We first prove the following three embeddings:

By the fact that \(\mathbb Q\) is homogeneous and universal for the class of finite orders.

Because the definition of \(\lt_{\mathbb Q}\) does not depend on the tree.

The ordering \(\lt_{\mathbb Q}\) on a perfect binary tree is a dense linear ordering.

Fix a coloring \(f:[\mathbb Q]^n\to k\).

Fix \(T\). The injection \(i\) induces \(f_T:\mathcal S_{2n-1}(T)\to K\) by \[f_T(S) = (f(i[H]))_{H\in[S]^{2n-1}}\]

WRONG: \(\quad f_T(\)\()=(f(q,q_0),f(q,q_1),f(q_0,q_1))\)

\(f_T(\)\()=(f(q,q_0),f(q,q_1),f(q_0,q_1), f(q_{00}, q_1), \dots)\)

Milliken's tree theorem gives us \(S\in\mathcal S_\omega(T)\) monochromatic for \(f_T\).

This subcopy of \(\mathbb Q\) uses at most \({2^{2n-1}-1}\choose n\) colors.

Let \(\sigma,\tau\in2^{<\omega}\).
We say that \(\sigma \mathbin{E_{\mathrm{pn}}}\tau\) if
\[|\sigma|\lt|\tau|\ \land\ \tau(|\sigma|)=1 \qquad \text{or} \qquad |\sigma|\gt|\tau|\ \land\ \sigma(|\tau|)=1 \]

Proof of the Rado graph theorem: Let \(T\) be a perfect binary tree and \(S\) be any strong subtree of \(T\). We first prove the following three embeddings:

By the fact that \(\mathbb G\) is homogeneous and universal for the class of finite graphs.

Because the definition of \(E_{\mathrm{pn}}\) does not depend on the tree.

Fix a coloring \(f:[\mathbb G]^n\to k\).

Fix \(T\). The injection \(i\) induces \(f_T:\mathcal S_{2n-1}(T)\to K\) by \[f_T(S) = (f(i[H]))_{H\in[S]^{2n-1}}\]

\(f_T(\)\()=(f(q,q_0),f(q,q_1),f(q_0,q_1), f(q_{00}, q_1),\dots)\)

Milliken's tree theorem gives us \(S\in\mathcal S_\omega(T)\) monochromatic for \(f_T\).

This subcopy of \(\mathbb G\) uses at most \({2^{2n-1}-1}\choose n\) colors.

Let \(\mathbb A\) be a structure on a language \(\mathcal L\). If one can prove that there exists a tree \(T\) and a set of relations \(R\) such that whenever \(S\in\mathcal S_\omega(T)\):

- \((T,R)\) embeds into \(\mathbb A\)
- True when \(\mathbb A\) homogeneous and universal for the finite substructures of \((T,R)\)
- \((S,R)\) embeds into \((T, R)\),
- True when \(R\) defined on \(2^{<\omega}\),
- \(\mathbb A\) embeds into \((S,R)\),
- ...

We now clearly see that understanding the computational strength of Milliken's tree theorem is essential to understand the computational strength of Devlin's theorem and the Rado graph theorem.

By \(\mathrm{MTT}^n\), we mean the statement of Milliken's tree theorem for strong subtrees of height \(n\).

\(\mathrm{MTT}^n\) implies \(\mathrm{RT}^n\), by applying the former to the coloring \(f'(S)=f(\{|\sigma|:\sigma\in S\})\).

Every computable instance of \(\mathrm{MTT}^n_k\) admits a \(\Delta^0_{2n-1}\) solution.

\(\forall F\in \mathcal S_{n-1}(T),\) \(\exists c_F\lt k,{\color{blue}\ell}\in\mathbb N,\) \(\forall {\color{brown}h}\gt\ell, \forall {\color{red}H}\subseteq T(h),\) \[F\cup H\in \mathcal S_{n}(T) \implies f(F\cup H)=c_F\]

\(\mathcal S_{\alpha}(T_0,\dots, T_{d-1})\) is the collection of all tuples:
\[(S_0,\dots,S_{d-1})\in\mathcal{S}_\alpha(T_0)\times\dots\times\mathcal S_\alpha(T_{d-1})\] with a common level function.

Let \(T_0,\dots,T_{d-1}\) be infinite trees with no leaves. For all \(k \geq 1\) and all \[f: \mathcal S_1(T_0, \cdots, T_{d-1}) \to k\] there exists \((S_0,\dots,S_{d-1}) \in \mathcal S_{\omega}(T_0,\dots,T_{d-1})\) such that \(f\) is constant on \(\mathcal S_1(S_0, \dots, S_{d-1})\).

\(T,f\) computable

- \(\mathrm{HL}\) is computably true.
- \(\mathrm{CMTT^n}\) admits \(\Delta^0_3\) solutions.

\(S\) arithmetical

\(S,f_{\mathrm{lim}}\) arithmetical

- \(\mathrm{MTT}^1\) is computably true (by \(\mathrm{HL}\)).
- Suppose \(\mathrm{MTT^{n-1}}\) admits arithmetical solutions.

\(S'\) arithmetical

\(\mathrm{MTT}^2\) does not imply \(\mathrm{ACA}_0\)

\(T,f\) cone avoiding

- \(\mathrm{HL}\) admits cone avoidance.
- \(\mathrm{CMTT^n}\) admits cone avoidance.

\(S,f\) cone avoiding

\(S\) cone avoiding,

\(f_{\mathrm{lim}}\) arithmetical in \(S\)

\(f_{\mathrm{lim}}\) arithmetical in \(S\)

\(\mathrm{HL}\) admits strong cone avoidance.

\(S'\) cone avoiding

\(\mathrm{RG}^n\) and \(\mathrm{DT}^n\) denote Rado graph theorem and Devlin's theorem with optimal bounds.

Recall that two strings define three levels: so we used \(\mathrm{MTT}^3\) for \(\mathrm{DT}^2\) and \(\mathrm{RG}^2\)

\(\mathrm{RG}^2\) admits cone avoidance, while \(\mathrm{DT}^2\) implies \(\mathrm{ACA}_0\)

\(\mathrm{MTT}^3_{\ell}\) and \(\mathrm{DT}^2_{\ell}\) denote the theorems when \(\ell\) colors are accepted in the solution.

\(\mathrm{MTT}^3_{2}\) and \(\mathrm{DT}^2_{4}\) admit cone avoidance, the bounds are optimal.

Thank you for your attention!