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Henry Towsner

Publications and source records attributed to Henry Towsner.

At least 55 records · Page 3Linked to original sources
arXiv2014

Separating Principles Below WKL0

In this paper, we study Ramsey-type Konig's Lemma, written RWKL, using a technique introduced by Lerman, Solomon, and the second author. This technique uses iterated forcing to construct an omega-model satisfying one principle T_1 but not another T_2. The technique often allows one to translate a "one step" construction (building an instance of T_2 along with a collection of solutions to each computable instance of T_1) into an omega-model separation (building a computable instance of T_2 together with a Turing ideal where T_1 holds). We illustrate this translation by separating d-DNR from DNR (reproving a result of Ambos-Spies, Kjos-Hanssen, Lempp, and Slaman), and then apply this technique to separate RWKL$ from DNR (which has been shown separately by Bienvenu, Patey, and Schafer).

math.LO↗
arXiv2014

Sigma-Algebras for Quasirandom Hypergraphs

We examine the correspondence between the various notions of quasirandomness for k-uniform hypergraphs and sigma-algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a sigma-algebra defined by a collection of subsets of [1,k]. We associate each notion of quasirandomness I with a collection of hypergraphs, the I-adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each I-adapted hypergraph. We then identify, for each I, a particular I-adapted hypergraph M_k[I] with the property that if G contains roughly the correct number of copies of M_k[I] then G is quasirandom in the sense of I. This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some notions of quasirandomness.

math.CO↗
arXiv2013

On Maximum Conservative Extensions

We investigate the set of Pi-1-2 sentences which are Pi-1-1 conservative over the theories of reverse mathematics RCA0+ISigma_n and ACA0. We exhibit new elements of these sets and conclude that the sets are Pi_2 complete. Along the way, we show that, over the theory RCA, induction for Sigma-n formulas has essentially no consequences for Delta-(n+1) formulas.

math.LO↗
arXiv2013

The Regularity Lemma with bounded VC Dimension

We give a proof of Szemeredi's regularity lemma in the special case of a graph with bounded VC dimension and show that it is possible to obtain "merely" doubly exponential bounds on the size of the partition in this case.

math.CO↗
arXiv2013

Separating principles below Ramsey's Theorem for Pairs

In recent years, there has been a substantial amount of work in reverse mathematics concerning natural mathematical principles that are provable from $\RT$, Ramsey's Theorem for Pairs. These principles tend to fall outside of the "big five" systems of reverse mathematics and a complicated picture of subsystems below $\RT$ has emerged. In this paper, we answer two open questions concerning these subsystems, specifically that $\ADS$ is not equivalent to $\CAC$ and that $\EM$ is not equivalent to $\RT$.

math.LO↗
arXiv2012

A Combinatorial Proof of the Dense Hindman Theorem

The Dense Hindman's Theorem states that, in any finite coloring of the integers, one may find a single color and a "dense" set $B_1$, for each $b_1\in B_1$ a "dense" set $B_2^{b_1}$ (depending on $b_1$), for each $b_2\in B_2^{b_1}$ a "dense" set $B_3^{b_1,b_2}$ (depending on $b_1,b_2$), and so on, such that for any such sequence of $b_i$, all finite sums belong to the chosen color. (Here density is often taken to be "piecewise syndetic", but the proof is unchanged for any notion of density satisfying certain properties.) This theorem is an example of a combinatorial statement for which the only known proof requires the use of ultrafilters or a similar infinitary formalism. Here we give a direct combinatorial proof of the theorem.

math.CO↗
arXiv2012

Partial Impredicativity in Reverse Mathematics

In reverse mathematics, is is possible to have a curious situation where we know that an implication does not reverse, but appear to have no information on on how to weaken the assumption while preserving the conclusion. A main cause of this phenomenon is the proof of a $Π^1_2$ sentence from the theory {\Pioo}. Using methods based on the functional interpretation, we introduce a family of weakenings of {\Pioo} and use them to give new upper bounds for the Nash-Williams Theorem of wqo theory and Menger's Theorem for countable graphs.

math.LO↗
arXiv2012

An approximate logic for measures

We present a logical framework for formalizing connections between finitary combinatorics and measure theory or ergodic theory that have appeared various places throughout the literature. We develop the basic syntax and semantics of this logic and give applications, showing that the method can express the classic Furstenberg correspondence and to give a short proof of the Szemerédi Regularity Lemma. We also derive some connections between the model-theoretic notion of stability and the Gowers uniformity norms from combinatorics.

math.LO↗
arXiv2012

Randomness and Non-ergodic Systems

We characterize the points that satisfy Birkhoff's ergodic theorem under certain computability conditions in terms of algorithmic randomness. First, we use the method of cutting and stacking to show that if an element x of the Cantor space is not Martin-Lof random, there is a computable measure-preserving transformation and a computable set that witness that x is not typical with respect to the ergodic theorem, which gives us the converse of a theorem by V'yugin. We further show that if x is weakly 2-random, then it satisfies the ergodic theorem for all computable measure-preserving transformations and all lower semi-computable functions.

math.LO↗
arXiv2011

Ultrafilters in Reverse Mathematics

We extend theories of reverse mathematics by a non-principal ultrafilter, and show that these are conservative extensions of the usual theories ACA0, ATR0, and Pi11-Comprehension.

math.LO↗
arXiv2011

A Simple Proof and Some Difficult Examples for Hindman's Theorem

We give a short, explicit proof of Hindman's Theorem that in every finite coloring of the integers, there is an infinite set all of whose finite sums have the same color. We give several exampls of colorings of the integers which do not have computable witnesses to Hindman's Theorem.

math.CO↗
arXiv2011

Transfinite Approximation of Hindman's Theorem

Hindman's Theorem states that in any finite coloring of the integers, there is an infinite set all of whose finite sums belong to the same color. This is much stronger than the corresponding finite form, stating that in any finite coloring of the integers there are arbitrarily long finite sets with the same property. We extend the finite form of Hindman's Theorem to a "transfinite" version for each countable ordinal, and show that Hindman's Theorem is equivalent to the appropriate transfinite approximation holding for every countable ordinal. We then give a proof of Hindman's Theorem by directly proving these transfinite approximations.

math.CO↗
arXiv2011

A Model Theoretic Proof of Szemerédi's Theorem

We present a short proof of Szemerédi's Theorem using a dynamical system enriched by ideas from model theory. The resulting proof contains features reminiscent of proofs based on both ergodic theory and on hypergraph regularity.

math.LO↗
arXiv2010

Metastability in the Furstenberg-Zimmer tower

According to the Furstenberg-Zimmer structure theorem, every measure-preserving system has a maximal distal factor, and is weak mixing relative to that factor. Furstenberg and Katznelson used this structural analysis of measure-preserving systems to provide a perspicuous proof of Szemerédi's theorem. Beleznay and Foreman showed that, in general, the transfinite construction of the maximal distal factor of a separable measure-preserving system can extend arbitrarily far into the countable ordinals. Here we show that the Furstenberg-Katznelson proof does not require the full strength of the maximal distal factor, in the sense that the proof only depends on a combinatorial weakening of its properties. We show that this combinatorially weaker property obtains fairly low in the transfinite construction, namely, by the $ω^{ω^ω}$th level.

math.DS↗
arXiv2009

Metastability and the Furstenberg-Zimmer Tower II: Polynomial and Multidimensional Szemeredi's Theorem

The Furstenberg-Zimmer structure theorem for $\mathbb{Z}^d$ actions says that every measure-preserving system can be decomposed into a tower of primitive extensions. Furstenberg and Katznelson used this analysis to prove the multidimensional Szemerédi's theorem, and Bergelson and Liebman further generalized to a polynomial Szemerédi's theorem. Beleznay and Foreman showed that, in general, this tower can have any countable height. Here we show that these proofs do not require the full height of this tower; we define a weaker combinatorial property which is sufficient for these proofs, and show that it always holds at fairly low levels in the transfinite construction (specifically, $ω^{ω^{ω^ω}}$).

math.DS↗