arXiv subjects

Henry Towsner

Publications and source records attributed to Henry Towsner.

At least 19 recordsLinked to original sources

arXiv2026

The finite big Ramsey degrees of Henson graphs are provable in $\mathrm{ACA}_0$

Let $\mathbb{H}_{n+1}$ denote a computable copy of the $(n+1)$-clique free universal homogeneous Henson graph, $G$ denote a finite subgraph of $\mathbb{H}_{n+1}$, and $k(G,n)$ denote the big Ramsey degree of $G$ in $\mathbb{H}_{n+1}$. We prove that for any computable coloring $\chi$ of the copies of $G$ in $\mathbb{H}_{n+1}$, there is a copy $\mathbb{H}'$ of $\mathbb{H}_{n+1}$ that is computable from $0^{(2\delta(G,n)-1)}$ in which $\chi$ takes no more than $k(G,n)$ colors, where $\delta(G,n)$ denotes the maximum number of levels of a diary for $G$ in $\mathbb{H}_{n+1}$ (this is a finite number). It follows that the statement, ``Henson graphs have finite big Ramsey degrees," is provable in ACA$_0'$. Combining this with a recent result of Cholak, Dobrinen, and McCoy \cite{CDM} yields the equivalence of the statement with ACA$_0'$ over RCA$_0$.

math.LO↗

arXiv2025

Higher-arity PAC learning, VC dimension and packing lemma

The aim of this note is to overview some of our work in Chernikov, Towsner'20 (arXiv:2010.00726) developing higher arity VC theory (VC$_n$ dimension), including a generalization of Haussler packing lemma, and an associated tame (slice-wise) hypergraph regularity lemma; and to demonstrate that it characterizes higher arity PAC learning (PAC$_n$ learning) in $n$-fold product spaces with respect to product measures introduced by Kobayashi, Kuriyama and Takeuchi'15. We also point out how some of the recent results in arXiv:2402.14294, arXiv:2505.15688, arXiv:2509.20404 follow from our work in arXiv:2010.00726.

stat.ML↗

arXiv2025

Averages of hypergraphs and higher arity stability

We show that $k$-ary functions giving the measure of the intersection of multi-parametric families of sets in probability spaces, e.g. $(x,y,z) \in X \times Y \times Z \mapsto \mu(P_{x,y} \cap Q_{x,z} \cap R_{y,z})$, satisfy a particularly strong form of hypergraph regularity. More generally, this applies to the (integral) averages of continuous combinations of functions of smaller arity. This result is connected to higher arity stability in model theory, that we discuss in the second part of the paper. We demonstrate that all hypergraphs embedding both into the half-simplex and into $GS(\mathbb{F}_3)$, the two known sources of failure of ternary stability, do satisfy an analogous regularity lemma -- hence strong ternary stability cannot be characterized simply by excluded hypergraphs.

math.CO↗

arXiv2025

Hanf Locality and Invariant Elementary Definability

We introduce some notions of invariant elementary definability which extend the notions of first-order order-invariant definability, and, more generally, definability invariant with respect to arbitrary numerical relations. In particular, we study invariance with respect to expansions which depend not only on (an ordering of) the universe of a structure, but also on the particular relations which determine the structure; we call such expansions \emph{presentations} of a structure. We establish two locality results in this context. The first is an extension of the original Hanf Locality Theorem to boolean queries which are invariantly definable over classes of locally finite structures with respect to \emph{elementary, neighborhood-bounded} presentations. The second is a non-uniform version of the Fagin-Stockmeyer-Vardi Hanf Threshold Locality Theorem to boolean queries which are invariantly definable over classes of bounded degree structures with respect to elementary, neighborhood-bounded, \emph{local} presentations.

math.LO↗

arXiv2025

Proofs that Modify Proofs, 1/2

This paper is a prelude and elaboration on Proofs that Modify Proofs. Here we present an ordinal analysis of a fragment of the $\mu$-calculus around the strength of parameter-free $\Pi^1_2$-comprehension using the same approach as that paper, interpreting functions on proofs as proofs in an expanded system. We build up the ordinal analysis in several stages, beginning by illustrating the method systems at the strength of paremeter-free $\Pi^1_1$-comprehension and full $\Pi^1_1$-comprehension.

math.LO↗

arXiv2025

The Complexity of the Set of Validities of a Theory

We study the collection of first-order logical schemata all of whose instances are theorems of a given theory $T$; we call these the validities of $T$ ($\mathsf{V}(T)$). It is easy to see that if $T$ is a decidable theory, then $\mathsf{V}(T)$ is distinct from the set of valid formulas of first-order logic as customarily understood. We provide a complete model-theoretic characterization of the complexity, in the sense of Turing degree, of $\mathsf{V}(T)$ for decidable theories $T$, and answer a question posed by Vaught in 1960 concerning the complexity of the collection of validities common to all decidable theories.

math.LO↗

arXiv2025

Polymorphic Ordinal Notations

We give an alternative presentation of the ordinal notation at the strength of $\Pi^1_1-CA_0$ which allows the "uncountable" notation $\Omega$ to be interpreted "polymorphically" - that is, we allow the notation to be interpreted as different cardinals depending on their context. This gives us a way to represent functions on ordinals within our ordinal notation system. We then use this idea to present an ordinal notation system for a system a bit weaker than parameter-free $\Pi^1_2-CA_0$.

math.LO↗

arXiv2024

A classification of incompleteness statements

For which choices of $X,Y,Z\in\{\Sigma^1_1,\Pi^1_1\}$ does no sufficiently strong $X$-sound and $Y$-definable extension theory prove its own $Z$-soundness? We give a complete answer, thereby delimiting the generalizations of G\"odel's second incompleteness theorem that hold within second-order arithmetic.

math.LO↗

arXiv2024

Intersecting sets in probability spaces and Shelah's classification

For $n \in \mathbb{N}$ and $\varepsilon > 0$, given a sufficiently long sequence of events in a probability space all of measure at least $\varepsilon$, some $n$ of them will have a common intersection. A more subtle pattern: for any $0 < p < q < 1$, we cannot find events $A_i$ and $B_i$ so that $\mu \left( A_i \cap B_j \right) \leq p$ and $\mu \left( A_j \cap B_i\right) \geq q$ for all $1 < i < j < n$, assuming $n$ is sufficiently large. This is closely connected to model-theoretic stability of probability algebras. We survey some results from our recent work on more complicated patterns that arise when our events are indexed by multiple indices. In particular, how such results are connected to higher arity generalizations of de Finetti's theorem in probability, structural Ramsey theory, hypergraph regularity in combinatorics, and model theory.

math.CO↗

arXiv2024

Proofs that Modify Proofs

In this paper we give an ordinal analysis of the theory of second order arithmetic. We do this by working with proof trees -- that is, "deductions" which may not be well-founded. Working in a suitable theory, we are able to represent functions on proof trees as yet further proof trees satisfying a suitable analog of well-foundedness. Iterating this process allows us to represent higher order functions as well: since functions on proof trees are just proof trees themselves, these functions can easily be extended to act on proof trees which are themselves understood as functions. The corresponding system of ordinals parallels this, using higher order collapsing function.

math.LO↗

arXiv2024

Perfect stable regularity lemma and slice-wise stable hypergraphs

We investigate various forms of (model-theoretic) stability for hypergraphs and their corresponding strengthenings of the hypergraph regularity lemma with respect to partitions of vertices. On the one hand, we provide a complete classification of the various possibilities in the ternary case. On the other hand, we provide an example of a family of slice-wise stable 3-hypergraphs so that for no partition of the vertices, any triple of parts has density close to 0 or 1. In particular, this addresses some questions and conjectures of Terry and Wolf. We work in the general measure theoretic context of graded probability spaces, so all our results apply both to measures in ultraproducts of finite graphs, leading to the aforementioned combinatorial applications, and to commuting definable Keisler measures, leading to applications in model theory.

math.CO↗

arXiv2023

From Saturated Embedding Tests to Explicit Algorithms

Quantifier elimination theorems show that each formula in a certain theory is equivalent to a formula of a specific form -- usually a quantifier-free one, sometimes in an extended language. Model theoretic embedding tests are a frequently used tool for proving such results without providing an explicit algorithm. We explain how proof mining methods can be adapted to apply to embedding tests, and provide two explicit examples, giving algorithms for theories of algebraic and real closed fields with a distinguished small subgroup corresponding to the embedding test proofs given by van den Dries and G\"unaydin.

math.LO↗

arXiv2023

An Aldous--Hoover Theorem for Radon Distributions

We show that the Aldous--Hoover Theorem, giving representations for exchangeable arrays of Borel-valued random variables, extends to random variables where the common distribution of the random variables is Radon, or even merely compact, a weaker condition that does not even require that the values come from a Hausdorff space. This extends work of Alam \cite{alam2023generalizing} who showed a similar generalization of the di Finetti--Hewitt-Savage Theorem.

math.PR↗

arXiv2023

Metric fixed point theory and partial impredicativity

We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in $\mathsf{RCA}_0$. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between $\mathsf{ATR}_0$ and $\Pi^1_1\mbox{-}\mathsf{CA}_0$. We also exhibit several weakenings of Caristi's theorem that are equivalent to $\mathsf{WKL}_0$ and to $\mathsf{ACA}_0$.

math.LO↗

arXiv2021

Borel combinatorics fail in HYP

We characterize the completely determined Borel subsets of HYP as exactly the omega_1^{ck} subsets of HYP. As a result, HYP believes there is a Borel well-ordering of the reals, that the Borel Dual Ramsey Theorem fails, and that every Borel d-regular bipartite graph has a Borel perfect matching, among other examples. Therefore, the Borel Dual Ramsey Theorem and several theorems of descriptive combinatorics are not theories of hyperarithmetic analysis. In the case of the Borel Dual Ramsey Theorem, this answers a question of Astor, Dzhafarov, Montalban, Solomon & the third author.

math.LO↗

arXiv2021

A Removal Lemma for Ordered Hypergraphs

We prove a removal lemma for induced ordered hypergraphs, simultaneously generalizing Alon--Ben-Eliezer--Fischer's removal lemma for ordered graphs and the induced hypergraph removal lemma. That is, we show that if an ordered hypergraph $(V,G,<)$ has few induced copies of a small ordered hypergraph $(W,H,\prec)$ then there is a small modification $G'$ so that $(V,G',<)$ has no induced copies of $(W,H,\prec)$. (Note that we do \emph{not} need to modify the ordering $<$.) We give our proof in the setting of an ultraproduct (that is, a Keisler graded probability space), where we can give an abstract formulation of hypergraph removal in terms of sequences of $\sigma$-algebras. We then show that ordered hypergraphs can be viewed as hypergraphs where we view the intervals as an additional notion of a ``very structured'' set. Along the way we give an explicit construction of the bijection between the ultraproduct limit object and the corresponding hyerpgraphon.

math.CO↗

arXiv2020

Hypergraph regularity and higher arity VC-dimension

We generalize the fact that graphs with small VC-dimension can be approximated by rectangles, showing that hypergraphs with small VC_k-dimension (equivalently, omitting a fixed finite (k+1)-partite (k+1)-uniform hypergraph) can be approximated by k-ary cylinder sets. In the language of hypergraph regularity, this shows that when H is a k'-uniform hypergraph with small VC_k-dimension for some k<k', the decomposition of H given by hypergraph regularity only needs the first k levels---one can approximate H using sets of vertices, sets of pairs, and so on up to sets of k-tuples---and that on most of the resulting k-ary cylinder sets, the density of H is either close to 0 or close to 1. We also show a suitable converse: k'-uniform hypergraphs with large VC_k-dimension cannot have such approximations uniformly under all measures on the vertices.

math.CO↗

arXiv2018

Explicit polynomial bounds on prime ideals in polynomial rings over fields

Suppose $I$ is an ideal of a polynomial ring over a field, $I\subseteq k[x_1,\ldots,x_n]$, and whenever $fg\in I$ with degree $\leq b$, then either $f\in I$ or $g\in I$. When $b$ is sufficiently large, it follows that $I$ is prime. Schmidt-G\"ottsch proved that "sufficiently large" can be taken to be a polynomial in the degree of generators of $I$ (with the degree of this polynomial depending on $n$). However Schmidt-G\"ottsch used model-theoretic methods to show this, and did not give any indication of how large the degree of this polynomial is. In this paper we obtain an explicit bound on $b$, polynomial in the degree of the generators of $I$. We also give a similar bound for detecting maximal ideals in $k[x_1,\ldots,x_n]$.

math.AC↗