A Ramsey theorem for multiposets

In the parlance of relational structures, the Finite Ramsey Theorem states that the class of all finite chains has the Ramsey property. A classical result from the 1980s claims that the class of all finite posets with a linear extension has the Ramsey property. In 2010 Sokić proved that the class of all finite structures consisting of several linear orders has the Ramsey property. This was followed by a 2017 result of Solecki and Zhao that the class of all finite posets with several linear extensions has the Ramsey property.Using the categorical reinterpretation of the Ramsey property in this paper we prove a common generalization of all these results. We consider multiposets to be structures consisting of several partial orders and several linear orders. We allow partial orders to extend each other in an arbitrary but fixed way, and require that every partial order is extended by at least one of the linear orders. We then show that the class of all finite multiposets conforming to a fixed template has the Ramsey property.     

On the Dual Ramsey Property for Finite Distributive Lattices

The class of finite distributive lattices, as many other natural classes of structures, does not have the Ramsey property. It is quite common, though, that after expanding the structures with appropriately chosen linear orders the resulting class has the Ramsey property. So, one might expect that a similar result holds for the class of all finite distributive lattices. Surprisingly, Kechris and Soki? have proved in 2012 that this is not the case: no expansion of the class of finite distributive lattices by linear orders satisfies the Ramsey property. In this paper we prove that the class of finite distributive lattices does not have the dual Ramsey property either. However, we are able to derive a dual Ramsey theorem for finite distributive lattices endowed with a particular linear order. Both results are consequences of the recently observed fact that categorical equivalence preserves the Ramsey property.     

The canonical Ramsey theorem and computability theory

Using the tools of computability theory and reverse mathematics, we study the complexity of two partition theorems, the Canonical Ramsey Theorem of Erdös and Rado, and the Regressive Function Theorem of Kanamori and McAloon. Our main aim is to analyze the complexity of the solutions to computable instances of these problems in terms of the Turing degrees and the arithmetical hierarchy. We succeed in giving a sharp characterization for the Canonical Ramsey Theorem for exponent 2 and for the Regressive Function Theorem for all exponents. These results rely heavily on a new, purely inductive, proof of the Canonical Ramsey Theorem. This study also unearths some interesting relationships between these two partition theorems, Ramsey's Theorem, and König's Lemma.

     

Transitive sets in Euclidean Ramsey theory

A finite set X in some Euclidean space Rn is called Ramsey if for any k there is a d such that whenever Rd is k-coloured it contains a monochromatic set congruent to X. This notion was introduced by Erd?s, Graham, Montgomery, Rothschild, Spencer and Straus, who asked if a set is Ramsey if and only if it is spherical, meaning that it lies on the surface of a sphere. This question (made into a conjecture by Graham) has dominated subsequent work in Euclidean Ramsey theory.In this paper we introduce a new conjecture regarding which sets are Ramsey; this is the first ever ‘rival’ conjecture to the conjecture above. Calling a finite set transitive if its symmetry group acts transitively—in other words, if all points of the set look the same—our conjecture is that the Ramsey sets are precisely the transitive sets, together with their subsets. One appealing feature of this conjecture is that it reduces (in one direction) to a purely combinatorial statement. We give this statement as well as several other related conjectures. We also prove the first non-trivial cases of the statement.Curiously, it is far from obvious that our new conjecture is genuinely different from the old. We show that they are indeed different by proving that not every spherical set embeds in a transitive set. This result may be of independent interest.     

Association schemes and hypergroups

In this paper, we investigate hypergroups which arise from association schemes in a canonical way; this class of hypergroups is called realizable. We first study basic algebraic properties of realizable hypergroups. Then we prove that two interesting classes of hypergroups (partition hypergroups and linearly ordered hypergroups) are realizable. Along the way, we prove that a certain class of projective geometries is equipped with a canonical association scheme structure which allows us to link three objects; association schemes, hypergroups, and projective geometries (see, Section 1.2 for details).     

Categorical equivalence and the Ramsey property for finite powers of a primal algebra

In this paper, we investigate the best known and most important example of a categorical equivalence in algebra, that between the variety of boolean algebras and any variety generated by a single primal algebra. We consider this equivalence in the context of Kechris-Pestov-Todor?evi? correspondence, a surprising correspondence between model theory, combinatorics and topological dynamics. We show that relevant combinatorial properties (such as the amalgamation property, Ramsey property and ordering property) carry over from a category to an equivalent category. We then use these results to show that the category whose objects are isomorphic copies of finite powers of a primal algebra \({\mathcal{A}}\) together with a particular linear ordering <, and whose morphisms are embeddings, is a Ramsey age (and hence a Fraïssé age). By the Kechris-Pestov-Todor?evi? correspondence, we then infer that the automorphism group of its Fraïssé limit is extremely amenable. This correspondence also enables us to compute the universal minimal flow of the Fraïssé limit of the class \({{\bf V}_{fin} \mathcal{(A)}}\) whose objects are isomorphic copies of finite powers of a primal algebra \({\mathcal{A}}\) and whose morphisms are embeddings.     

Monotone meta-Lindelöf spaces

In this paper, we study the monotone meta-Lindelöf property. Relationships between monotone meta-Lindelöf spaces and other spaces are investigated. Behaviors of monotone meta-Lindelöf GO-spaces in their linearly ordered extensions are revealed.     

Automorphism Groups of Countably Categorical Linear Orders are Extremely Amenable

We show that the automorphism groups of countably categorical linear orders are extremely amenable. Using methods of Kechris, Pestov, and Todorcevic, we use this fact to derive a structural Ramsey theorem for certain families of finite ordered structures with finitely many partial equivalence relations with convex classes.     

Pairs,sets and sequences in first-order theories

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is definitionally equivalent to an extension in the same language of a system of weak set theory called WS.      

The shift graph and the Ramsey degree of {[\mathbb{N}]}^{\omega}

We address several questions related to the Ramsey degree of the class of infinite structures isomorphic to the positive integers with the usual ordering. In particular we study the strength of stating that this class has finite Ramsey degree in the absence of the axiom of choice. We show that if the shift graph has finite chromatic number then this class has infinite Ramsey degree.     

On the Ramsey Number of Sparse 3-Graphs

We consider a hypergraph generalization of a conjecture of Burr and Erd?s concerning the Ramsey number of graphs with bounded degree. It was shown by Chvátal, Rödl, Trotter, and Szemerédi [The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), no. 3, 239–243] that the Ramsey number R(G) of a graph G of bounded maximum degree is linear in |V(G)|. We derive the analogous result for 3-uniform hypergraphs.     

Ramsey goodness and beyond

In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five — all but one — of the Burr-Erdős problems.     

The finite model property for semilinear substructural logics

In this paper, we show that the finite model property fails for certain non‐integral semilinear substructural logics including Metcalfe and Montagna's uninorm logic and involutive uninorm logic, and a suitable extension of Metcalfe, Olivetti and Gabbay's pseudo‐uninorm logic. Algebraically, the results show that certain classes of bounded residuated lattices that are generated as varieties by their linearly ordered members are not generated as varieties by their finite members.     

κ-compactness,extent and the Lindelöf number in LOTS

We study the behaviour of ?-compactness, extent and Lindelöf number in lexicographic products of linearly ordered spaces. It is seen, in particular, that for the case that all spaces are bounded all these properties behave very well when taking lexicographic products. We also give characterizations of these notions for generalized ordered spaces.     

Interpretability degrees of finitely axiomatized sequential theories

In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory—like Elementary Arithmetic EA, IΣ₁, or the Gödel–Bernays theory of sets and classes GB—have suprema. This partially answers a question posed by ?vejdar in his paper (Commentationes Mathematicae Universitatis Carolinae 19:789–813, 1978). The partial solution of ?vejdar’s problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree structure of the degrees of all finitely axiomatized sequential theories. In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result.     

Coloured solutions of equations in finite groups

In this article, we consider the relations between colourings and some equations in finite groups. We will express relations linking the numbers of the differently coloured solutions of an equation that depend only on the cardinality of the colouring and not on the distribution of the colours. This gives a link between Ramsey theory that investigates the existence of monochromatic solutions and what is now called anti-Ramsey theory that investigates the existence of rainbow solutions. Both theories are in expansion. We will apply these results to the counting of rainbow 3-term arithmetic progressions in any abelian group with equinumerous three-colouring and to the counting of points on a conic defined on a finite field. We will end by discussing the generalised case of a system of equations.     

Borsuk's problem and the chromatic numbers of metric spaces

In this paper, we will give a short survey of various old and new results concerning two closely connected problems of combinatorial geometry - that of K. Borsuk and that of E. Nelson - P. Erdös - H. Hadwiger. We will also present here our very recent achievements on Ramsey numbers.     

Ideal version of Ramsey’s theorem

We consider various forms of Ramsey’s theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem which are connected with ideals of subsets of natural numbers. We characterize ideals with properties considered. We show that, in a sense, Ramsey’s theorem, the monotone subsequence theorem and the Bolzano-Weierstrass theorem characterize the same class of ideals. We use our results to show some versions of density Ramsey’s theorem (these are similar to generalizations shown in [P. Frankl, R. L. Graham, and V. Rödl: Iterated combinatorial density theorems.     

Degrees of asynchronously automaton transformations

In this paper we study a partially ordered set of degrees of asynchronously automaton transformations. We prove the existence of a continuum of atoms in this set, the embeddability of every finite linearly ordered set as an initial segment in it, and the nonextensibility of an embedding of partially ordered sets.