arXiv subjects

Dragan Mašulović

Publications and source records attributed to Dragan Mašulović.

At least 19 recordsLinked to original sources

arXiv2026

Hall's universal group does not have finite big Ramsey degrees

In this paper we show that the Hall's universal group does not have finite big Ramsey degrees. Our strategy consists of piggybacking on the recent result of Hubička, Konečný, Todorčević and Zucker (announced at EUROCOMB 2025) that the random edge-labelled graph (the Fraïssé limit of the class of all finite complete edge-labelled graphs where the set of labels is countably infinite) does not have finite big Ramsey degrees. We then use the machinery of category theory to transport this result from the context of edge-labelled graphs to the context of groups. The main step in the process is the construction of a functor from the category of edge-labelled graphs and embeddings to the category of groups and group embeddings which takes finite graphs to finite groups. This makes is possible for us to build a subgroup of the Hall's universal group which encodes the random edge-labelled graph. Our main result shows that several families finite groups, including non-cyclic abelian groups, do not have finite big Ramsey degrees in the Hall's universal group. Actually, we show that all potential candidates for groups with finite big Ramsey degrees have to be solvable.

math.CO↗

arXiv2026

Fraïssé's Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition

In this paper we show that a countable structure admitting a finite monomorphic decomposition has finite big Ramsey degrees if and only if so does every monomorphic part in its minimal monomorphic decomposition. The necessary prerequisite for this result is the characterization of monomorphic structures with finite big Ramsey degrees: a countable monomorphic structure has finite big Ramsey degrees if and only if it is chainable by a chain with finite big Ramsey degrees. Interestingly, both characterizations require deep structural properties of chains. Fraïssé's Conjecture (actually, its positive resolution due to Laver) is instrumental in the characterization of monomorphic structures with finite big Ramsey degrees, while the analysis of big Ramsey combinatorics of structures admitting a finite monomorphic decomposition requires a product Ramsey theorem for big Ramsey degrees. We find this last result particularly intriguing because big Ramsey degrees misbehave notoriously when it comes to general product statements. In the addendum, we combine a recent result by Oudrar and Pouzet with our analysis of finite big Ramsey degrees for structures admitting finite monomorphic decomposition to characterize the existence of finite Big Ramsey degrees for all countable relational structures whose language has a linear order and age has polynomial growth.

math.LO↗

arXiv2026

Corrigendum and Addendum to "Fra\"ıssé's Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition''

In Section 6 of the paper ``Fra\"ıssé's Conjecture and big Ramsey degrees of structures admitting finite monomorphic decomposition'', we applied the methods developed in earlier sections to show that a certain reduct of the generic permutation has finite big Ramsey degrees. Unfortunately, this reduct was incorrectly identified as the generic partial order. We are grateful to Jan Hubička for bringing this error to our attention. In this note we correct the statements that rely on this misidentification and demonstrate that the reduct in question is in fact the generic 2-dimensional partial order. We emphasize that the arguments presented in Section 6 remain valid, with the sole exception of the Claim in the proof of Theorem 6.4, whose role was to (incorrectly) identify the reduct of the generic permutation as the generic partial order. This correction has an unexpected positive consequence. Rather than reproving a well-known result whose existing proof is already notably elegant, this note demonstrates that our general framework can be used to establish that a previously unexplored class of generic relational structures has finite big Ramsey degrees. This observation opens a potentially new direction for further research in the thriving area of big Ramsey combinatorics. In the addendum, we combine a recent result by Oudrar and Pouzet with our analysis of finite big Ramsey degrees for structures admitting finite monomorphic decomposition to characterize the existence of finite Big Ramsey degrees for all countable relational structures whose language has a linear order and age has polynomial growth.

math.CO↗

arXiv2026

Deducibility in Sudoku

In this paper we provide a formalism, Sudoku logic, in which a solution is logically deducible if for every cell of the grid we can provably exclude all but a single option. We prove that the deductive system of Sudoku logic is sound and complete, and as a consequence of that we prove that a Sudoku puzzle has a unique solution if and only if it has a deducible solution. Using the classification of fundamental Sudoku transformations by Adler and Adler we then formalize the notion of symmetry in Sudoku and provide a formal proof of Gurth's Symmetrical Placement Theorem. In the concluding section we present a Sudoku formula that captures the idea of a Sudoku grid having a unique solution. It turns out that this formula is an axiom of Sudoku logic, making it possible for us to offer to the Sudoku community a resolution of the Uniqueness Controversy: if we accept Sudoku logic as presented in this paper, there is no controversy! Uniqueness is an axiom and, as any other axiom, may freely be used in any Sudoku deduction.

math.LO↗

arXiv2026

Coalgebras for categorical deep learning: Representability and universal approximation

Categorical deep learning (CDL) has recently emerged as a framework that leverages category theory to unify diverse neural architectures. While geometric deep learning (GDL) is grounded in the specific context of invariants of group actions, CDL aims to provide domain-independent abstractions for reasoning about models and their properties. In this paper, we contribute to this program by developing a coalgebraic foundation for equivariant representation in deep learning, as classical notions of group actions and equivariant maps are naturally generalized by the coalgebraic formalism. Our first main result demonstrates that, given an embedding of data sets formalized as a functor from SET to VECT, and given a notion of invariant behavior on data sets modeled by an endofunctor on SET, there is a corresponding endofunctor on VECT that is compatible with the embedding in the sense that this lifted functor recovers the analogous notion of invariant behavior on the embedded data. Building on this foundation, we then establish a universal approximation theorem for equivariant maps in this generalized setting. We show that continuous equivariant functions can be approximated within our coalgebraic framework for a broad class of symmetries. This work thus provides a categorical bridge between the abstract specification of invariant behavior and its concrete realization in neural architectures.

cs.LG↗

arXiv2026

Big Ramsey combinatorics of the Cantor set and a simple proof of Blass' perfect set theorem

In this paper we present a simple approach to big Ramsey combinatorics of the Cantor set $2^ω$. Using Infinite Dual Ramsey Theorem of Carlson and Simpson, we show that $2^ω$, viewed as a topological space, has finite big Ramsey degrees. We then examine several natural topological first-order structures arising from the Cantor set and prove that each of them inherits finite big Ramsey degrees. As a consequence, we obtain a simple proof of Blass' perfect set theorem, although our method does not recover the sharp bound $(n-1)!$ for the number of colors. We also show that the complete Boolean algebra on countably many atoms has finite big Ramsey degrees, in contrast with the recent result showing that the countable atomless Boolean algebra does not have big Ramsey degrees.

math.LO↗

arXiv2025

Ramsey degrees: big v. small

In this paper we investigate algebraic properties of big Ramsey degrees in categories satisfying some mild conditions. As the first nontrivial consequence of the generalization we advocate in this paper we prove that small Ramsey degrees are the minima of the corresponding big ones. We also prove that big Ramsey degrees are subadditive and show that equality is enforced by an abstract property of objects we refer to as self-similarity. Finally, we apply the abstract machinery developed in the paper to show that if a countable relational structure has finite big Ramsey degrees, then so do its quantifier-free reducts. In particular, it follows that the reducts of (Q, <), the random graph, the random tournament and (Q, <, 0) all have finite big Ramsey degrees.

math.CO↗

arXiv2025

Classes of finite relational structures over finite languages have dual Ramsey degrees

Classical Ramsey theory has successfully extended to relational structures, yielding a wealth of results that have profoundly influenced other areas of mathematics. Interestingly, the same development has not occurred in the case of dual Ramsey theory. The main goal of this paper is to advance the dual Ramsey theory for finite relational structures with respect to natural structure-preserving maps. Tools from category theory prove instrumental in this endeavor, as was previously the case for finite algebraic systems where the dual Ramsey property had been established for every class of finite algebras coming from an equationally defined class. One cannot help but feel that dual Ramsey phenomena are deeply connected to categorical strategies.

math.CO↗

arXiv2024

On Ramsey degrees, compactness and approximability

One of the consequences of the Compactness Principle in structural Ramsey theory is that the small Ramsey degrees cannot exceed the corresponding big Ramsey degrees, thereby justifying the choice of adjectives. However, it is unclear what happens in the realm of dual Ramsey degrees due to the lack of the compactness argument that applies to that setting. In this paper we present a framework within which both "direct" and dual Ramsey statements can be stated and reasoned about in a uniform fashion. We introduce the notion of approximability which yields a general compactness argument powerful enough to prove statements about both "direct" and dual Ramsey phenomena. We conclude the paper with an application of the new strategies by generalizing Voigt's $\star$-version of the Infinite Ramsey Theorem to a large class of relational structures and deriving a Ramsey statement for "loose colorings" of enumerated Fra\"ıssé limits.

math.LO↗

arXiv2024

Coalgebraic methods for Ramsey degrees of unary algebras

In this paper we are interested in the existence of small and big Ramsey degrees of classes of finite unary algebras in arbitrary (not necessarily finite) algebraic language $Ω$. We think of unary algebras as $M$-sets where $M = Ω^*$ is the free monoid of words over the alphabet $Ω$ and show that for an arbitrary monoid $M$ (finite or infinite) the class of all finite $M$-sets has finite small Ramsey degrees. This immediately implies that the class of all finite $G$-sets, where $G$ is an arbitrary group (finite or infinite), has finite small Ramsey degrees, and that the class of all finite unary algebras over an arbitrary (finite or infinite) algebraic language $Ω$ has finite small Ramsey degrees. This generalizes some Ramsey-type results of M.\ Sokić concerning finite unary algebras over finite languages and finite $G$-sets for finite groups~$G$. To do so we develop a completely new strategy that relies on the fact that right adjoints preserve the Ramsey property. We then treat $M$-sets as Eilenberg-Moore coalgebras for "half a comonad" and using pre-adjunctions transport the Ramsey properties we are interested in from the category of finite or countably infinite chains of order type $ω$. Moreover, we show that finite objects have finite big Ramsey degrees in the corresponding cofree structures over countably many generators.

math.CO↗

arXiv2024

On $k$-ary parts of maximal clones

The main problem of clone theory is to describe the clone lattice for a given basic set. For a two-element basic set this was resolved by E.L. Post, but for at least three-element basic set the full structure of the lattice is still unknown, and the complete description in general is considered to be hopeless. Therefore, it is studied by its substructures and its approximations. One of the possible directions is to examine $k$-ary parts of the clones and their mutual inclusions. In this paper we study $k$-ary parts of maximal clones, for $k\geq2$, building on the already known results for their unary parts. It turns out that the poset of $k$-ary parts of maximal clones defined by central relations contains long chains.

math.RA↗

arXiv2023

Tukey reducibility for categories -- In search of the strongest statement in finite Ramsey theory

Every statement of the Ramsey theory of finite structures corresponds to the fact that a particular category has the Ramsey property. We can, then, compare the strength of Ramsey statements by comparing the ``Ramsey strength'' of the corresponding categories. The main thesis of this paper is that establishing pre-adjunctions between pairs of categories is an appropriate way of comparing their ``Ramsey strength''. What comes as a pleasant surprise is that pre-adjunctions generalize the Tukey reducibility in the same way categories generalize preorders. In this paper we set forth a classification program of statements of finite Ramsey theory based on their relationship with respect to this generalized notion of Tukey reducibility for categories. After identifying the ``weakest'' Ramsey category, we prove that the Finite Dual Ramsey Theorem is as powerful as the full-blown version of the Graham-Rothschild Theorem, and conclude the paper with the hypothesis that the Finite Dual Ramsey Theorem is the ``strongest'' of all finite Ramsey statements.

math.CT↗

arXiv2023

A short note on the characterization of countable chains with finite big Ramsey spectra

In this short note we confirm the deep structural correspondence between the complexity of a countable scattered chain (= strict linear order) and its big Ramsey combinatorics: we show that a countable scattered chain has finite big Ramsey degrees if and only if it is of finite Hausdorff rank. This also provides a complete characterization of countable chains whose big Ramsey spectra are finite. We expand the notion of big Ramsey spectrum to monomorphic structures and give a sufficient condition for a monomorphic countable structure to have finite big Ramsey spectrum.

math.LO↗

arXiv2023

Ramsey properties of products and pullbacks of categories and the Grothendieck construction

In this paper we provide purely categorical proofs of two important results of structural Ramsey theory: the result of M.\ Sokić that the free product of Ramsey classes is a Ramsey class and the result of M.\ Bodirsky, M.\Pinsker and T.\ Tsankov that adding constants to the language of a Ramsey class preserves the Ramsey property. The proofs that we present here ignore the model-theoretic background of these statements. Instead, they focus on categorical constructions by which the classes can be constructed, generalizing the original statements along the way. It turns out that the restriction to classes of relational structures, although fundamental for the original proof strategies, is not relevant for the statements themselves. The categorical proofs we present here remove all the restrictions on the signature of first-order structures and provide the information not only about the Ramsey property but also about the Ramsey degrees.

math.CT↗

arXiv2022

From Ramsey degrees to Ramsey expansions via weak amalgamation

Under no additional assumptions, in this paper we construct a Ramsey expansion for every category of finite objects with finite small Ramsey degrees. Our construction is based on the relationship between small Ramsey degrees, weak amalgamation and recent results about weak Fraïssé categories. Namely, generalizing the fact that every Ramsey class has amalgamation, we show that classes with finite Ramsey degrees have weak amalgamation. We then invoke the machinery of weak Fraïssé categories to perform the construction. This improves previous similar results where an analogous construction was carried out under the assumption that everything sits comfortably in a bigger class with enough infrastructure, and that in this wider context there is an ultrahomogeneous structure under whose umbrella the construction takes place.

math.LO↗

arXiv2022

Ramsey degrees and entropy of combinatorial structures

Close connections between various notions of entropy and the apparatus of category theory have been observed already in the 1980s and more vigorously developed in the past ten years. The starting point of the paper is the recent categorical understanding of structural Ramsey degrees, which then leads to a way to compute entropy of an object in a small category not as a measure of statistical, but as a measure of its combinatorial complexity. The new entropy function we propose, the Ramsey entropy, is a real-valued invariant of an object in an arbitrary small category. We require no additional categorical machinery to introduce and prove the properties of this entropy. Motivated by combinatorial phenomena (structural Ramsey degrees) we build the necessary infrastructure and prove the fundamental properties using only special partitions imposed on homsets. We conclude the paper with the discussion of the maximal Ramsey entropy on a category that we refer to as the Ramsey-Boltzmann entropy.

math.CO↗

arXiv2021

Big Ramsey spectra of countable chains

A big Ramsey spectrum of a countable chain (i.e. strict linear order) C is a sequence of big Ramsey degrees of finite chains computed in C. In this paper we consider big Ramsey spectra of countable scattered chains. We prove that countable scattered chains of infinite Hausdorff rank do not have finite big Ramsey spectra, and that countable scattered chains of finite Hausdorff rank with bounded finite sums have finite big Ramsey spectra. Since big Ramsey spectra of all non-scattered countable chains are finite by results of Galvin, Laver and Devlin, in order to complete the characterization of countable chains with finite big Ramsey spectra (or degrees) one still has to resolve the remaining case of countable scattered chains of finite Hausdorff rank whose finite sums are not bounded.

math.CO↗

arXiv2020

The Kechris-Pestov-Todor\vcević correspondence from the point of view of category theory

The Kechris-Pestov-Todorčević correspondence (KPT-correspondence for short) is a surprising correspondence between model theory, combinatorics and topological dynamics. In this paper we present a categorical re-interpretation of (a part of) the KPT-correspondence with the aim of proving a dual statement. Our strategy is to take a "direct" result and then analyze the necessary infrastructure that makes the result true by providing a purely categorical proof of the categorical version of the result. We can then capitalize on the Duality Principle to obtain the dual statements almost for free. We believe that the dual version of the KPT-correspondence can not only provide the new insights into the interplay of combinatorial, model-theoretic and topological phenomena this correspondence binds together, but also explores the limits to which categorical treatment of combinatorial phenomena can take us.

math.CT↗