Open questions about Ramsey-type statements in reverse mathematics

Ramsey's theorem states that for any coloring of the n-element subsets of N with finitely many colors, there is an infinite set H such that all n-element subsets of H have the same color. The strength of consequences of Ramsey's theorem has been extensively studied in reverse mathematics and under various reducibilities, namely, computable reducibility and uniform reducibility. Our understanding of the combinatorics of Ramsey's theorem and its consequences has been greatly improved over the past decades. In this paper, we state some questions which naturally arose during this study. The inability to answer those questions reveals some gaps in our understanding of the combinatorics of Ramsey's theorem.Comment: 15 page

What is good mathematics?

Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer\'edi's theorem is presented.Comment: 12 pages, no figures. To appear, Bull. Amer. Math. So

Spurious, Emergent Laws in Number Worlds

We study some aspects of the emergence of logos from chaos on a basal model of the universe using methods and techniques from algorithmic information and Ramsey theories. Thereby an intrinsic and unusual mixture of meaningful and spurious, emerging laws surfaces. The spurious, emergent laws abound, they can be found almost everywhere. In accord with the ancient Greek theogony one could say that logos, the Gods and the laws of the universe, originate from "the void," or from chaos, a picture which supports the unresolvable/irreducible lawless hypothesis. The analysis presented in this paper suggests that the "laws" discovered in science correspond merely to syntactical correlations, are local and not universal.Comment: 24 pages, invited contribution to "Contemporary Natural Philosophy and Philosophies - Part 2" - Special Issue of the journal Philosophie

Ramsey-type graph coloring and diagonal non-computability

A function is diagonally non-computable (d.n.c.) if it diagonalizes against the universal partial computable function. D.n.c. functions play a central role in algorithmic randomness and reverse mathematics. Flood and Towsner asked for which functions h, the principle stating the existence of an h-bounded d.n.c. function (DNR_h) implies the Ramsey-type K\"onig's lemma (RWKL). In this paper, we prove that for every computable order h, there exists an~$\omega$-model of DNR_h which is not a not model of the Ramsey-type graph coloring principle for two colors (RCOLOR2) and therefore not a model of RWKL. The proof combines bushy tree forcing and a technique introduced by Lerman, Solomon and Towsner to transform a computable non-reducibility into a separation over omega-models.Comment: 18 page

The Deluge of Spurious Correlations in Big Data

International audienceVery large databases are a ma jor opp ortunity for science and data analytics is a remarkable new field of investigation in computer science. The effectiveness of these toolsis used to support a “philosophy” against the scientific method as developed throughout history. According to this view, computer-discovered correlations should replace understanding and guide prediction and action. Consequently, there will be no need to givescientific meaning to phenomena, by proposing, say, causal relations, since regularities in very large databases are enough: “with enough data, the numbers speak for themselves”. The “end of science” is proclaimed. Using classical results from ergodic theory, Ramsey theory and algorithmic information theory, we show that this “philosophy” is wrong. For example, we prove that very large databases have to contain arbitrary correlations. These correlations appear only due to the size, not the nature, of data. They can be found in “randomly” generated, large enough databases, which - as we will prove - implies that most correlations are spurious. Too much information tends to behave like very little information. The scientific method can be enriched by computer mining in immense databases, but not replaced by it

Graph removal lemmas

The graph removal lemma states that any graph on n vertices with o(n^{v(H)}) copies of a fixed graph H may be made H-free by removing o(n^2) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and computer science. In this survey we discuss these lemmas, focusing in particular on recent improvements to their quantitative aspects.Comment: 35 page

A Time Hierarchy Theorem for the LOCAL Model

The celebrated Time Hierarchy Theorem for Turing machines states, informally, that more problems can be solved given more time. The extent to which a time hierarchy-type theorem holds in the distributed LOCAL model has been open for many years. It is consistent with previous results that all natural problems in the LOCAL model can be classified according to a small constant number of complexities, such as $O(1),O(\log^* n), O(\log n), 2^{O(\sqrt{\log n})}$, etc. In this paper we establish the first time hierarchy theorem for the LOCAL model and prove that several gaps exist in the LOCAL time hierarchy. 1. We define an infinite set of simple coloring problems called Hierarchical $2\frac{1}{2}$-Coloring}. A correctly colored graph can be confirmed by simply checking the neighborhood of each vertex, so this problem fits into the class of locally checkable labeling (LCL) problems. However, the complexity of the $k$-level Hierarchical $2\frac{1}{2}$-Coloring problem is $\Theta(n^{1/k})$, for $k\in\mathbb{Z}^+$. The upper and lower bounds hold for both general graphs and trees, and for both randomized and deterministic algorithms. 2. Consider any LCL problem on bounded degree trees. We prove an automatic-speedup theorem that states that any randomized $n^{o(1)}$-time algorithm solving the LCL can be transformed into a deterministic $O(\log n)$-time algorithm. Together with a previous result, this establishes that on trees, there are no natural deterministic complexities in the ranges $\omega(\log^* n)$---$o(\log n)$ or $\omega(\log n)$---$n^{o(1)}$. 3. We expose a gap in the randomized time hierarchy on general graphs. Any randomized algorithm that solves an LCL problem in sublogarithmic time can be sped up to run in $O(T_{LLL})$ time, which is the complexity of the distributed Lovasz local lemma problem, currently known to be $\Omega(\log\log n)$ and $O(\log n)$

Computability Theory (hybrid meeting)

Over the last decade computability theory has seen many new and fascinating developments that have linked the subject much closer to other mathematical disciplines inside and outside of logic. This includes, for instance, work on enumeration degrees that has revealed deep and surprising relations to general topology, the work on algorithmic randomness that is closely tied to symbolic dynamics and geometric measure theory. Inside logic there are connections to model theory, set theory, effective descriptive set theory, computable analysis and reverse mathematics. In some of these cases the bridges to seemingly distant mathematical fields have yielded completely new proofs or even solutions of open problems in the respective fields. Thus, over the last decade, computability theory has formed vibrant and beneficial interactions with other mathematical fields. The goal of this workshop was to bring together researchers representing different aspects of computability theory to discuss recent advances, and to stimulate future work