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

Plan of the talk

  1. The combinatorial relations between
    • Devlin's theorem,
    • The Rado graph theorem
    • Milliken's tree theorem
  2. The computable strength of Milliken's Tree Theorem and two applications
 Introduction to Reverse Mathematics
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 \(A\).

The computability-theoretic perspective

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

What about separations?

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 \(\mathrm{ACA}_0\).
 Ramsey's theorem on structures

Adding structure to the sets of 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 \([S]^n\).
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 \([\mathbb S]^n\).

  • 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.

Structures with finite big Ramsey numbers

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

DLO

Dense Linear Order
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 \(\ell_1=1\), \(\ell_2=2\), \(\ell_3=16\), \(\ell_4=272\), ...

Rado graph

The Rado graph
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.

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 \(T\)}\]
\(\mathcal S_\alpha(T)\) is the set of all strong subtrees of \(T\) of height \(\alpha\).
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 \(\mathcal S_n(S)\).

How Milliken implies Devlin and Rado

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

Devlin

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.

Rado graph

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 \((S,R)\),
    • ...
Then by Milliken's tree theorem, \(\mathbb A\) has finite big Ramsey numbers.

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.

The computational strength of Milliken's tree 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 \(S\)
\(\mathrm{HL}\) 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!