arXiv subjects

Bertalan Bodor

Publications and source records attributed to Bertalan Bodor.

16 recordsLinked to original sources

arXiv2026

Equivalences of promise compactness principles

For a pair of finite relational structures $(\mathfrak{A},\mathfrak{B})$ such that $\mathfrak{A}$ homomorphically maps to $\mathfrak{B}$ we denote by $K_{(\mathfrak{A},\mathfrak{B})}$ the following statement: for all structures $\mathfrak{I}$ with the same signature as $\mathfrak{A}$ if all finite substructures of $\mathfrak{I}$ homomorphically maps to $\mathfrak{A}$ then $\mathfrak{I}$ homomorphically maps to $\mathfrak{B}$. In this article, we show that if $(\mathfrak{A},\mathfrak{B})$ has no Ol\v{s}\'{a}k polymorphism, then $K_{(\mathfrak{A},\mathfrak{B})}$ is equivalent to the ultrafilter principle over $\operatorname{ZF}$. This includes the statements $K_{(K_3,K_5)}$ and $K_{(H_2,H_c)}$ for all $c\geq 2$ where $K_n$ denotes the clique of size $n$ and $H_k$ denotes the ternary not-all-equal structure on a $k$-element set. This means, for example, that in any $\operatorname{ZF}$ model, if every finitely 3-colourable graph can be coloured by 5 colours then all these graphs can in fact be coloured by 3 colours.

math.CO↗

arXiv2025

Taking model-complete cores

A first-order theory $T$ is a model-complete core theory if every first-order formula is equivalent modulo $T$ to an existential positive formula; a core companion of a theory $T$ is a model-complete core theory $S$ such that every model of $T$ maps homomorphically to a model of $S$ and vice-versa. Whilst core companions may not exist in general, if they exist, they are unique. Moreover, $\omega$-categorical theories always have a core companion, which is also $\omega$-categorical. We show that many model-theoretic properties, such as stability, $\mathrm{NIP}$, simplicity, and $\mathrm{NSOP}_k$ for ${k\in\mathbb{N}_{>0}}$, are preserved by moving to the core companion of a complete theory. On the other hand, we show that the classes of theories of structures interpretable over $({\mathbb N};=)$ and over $({\mathbb Q};<)$ are both not closed under taking core companions. The first class is contained in the class of theories of $\omega$-stable first-order reducts of finitely homogeneous relational structures, which was studied by Lachlan in the 80's. We conjecture the two classes to be equal. To support our conjecture we prove that all structures in Lachlan's class are trace definable in $(\mathbb{N}; =)$, confirming a conjecture of Walsberg.

math.LO↗

arXiv2025

Labelled growth rates of $\omega$-categorical structures and applications in choiceless set theory

We study the labelled growth rate of an $\omega$-categorical structure $\mathfrak{A}$, i.e., the number of orbits of $Aut(\mathfrak{A})$ on $n$-tuples of distinct elements, and show that the model-theoretic property of monadic stability yields a gap in the spectrum of allowable labelled growth rates. As a further application, we obtain gap in the spectrum of allowable labelled growth rates in hereditary graph classes, with no a priori assumption of $\omega$-categoricity. We also establish a way to translate results about labelled growth rates of $\omega$-categorical structures into combinatorial statements about sets with weak finiteness properties in the absence of the axiom of choice, and derive several results from this translation.

math.LO↗

arXiv2025

Structures with not too fast unlabelled growth

Let $\mathscr{S}$ be the class of all structures whose growth rate on orbits of subsets of size $n$ is not faster than $\frac{2^n}{p(n)}$ for any polynomial $p$. In this article we give a complete classification of all structures in $\mathscr{S}$ in terms of their automorphism groups. As a consequence of our classification we show that $\mathscr{S}$ has only countably many structures up to bidefinability, all these structures are first-order interpretable in $(\mathbb{Q};<)$ and they are interdefinable with a finitely bounded homogeneous structure. Furthermore, we also show that all structures in $\mathscr{S}$ have finitely many first-order reduct up to interdefinability, thereby confirming Thomas' conjecture for the class $\mathscr{S}$.

math.LO↗

arXiv2025

Structures preserved by primitive actions of $S_\omega$

We present a dichotomy for structures $A$ that are preserved by primitive actions of $S_{\omega} = \text{Sym}({\mathbb N})$: such a structure primitively positively constructs all finite structures and the constraint satisfaction problem is NP-complete, or the constraint satisfaction problem for $A$ is in P. To prove our result, we study the first-order reducts of the Johnson graph $J(k)$, for $k \geq 2$, whose automorphism group $G$ equals the action of $\text{Sym}({\mathbb N})$ on the set $V$ of $k$-element subsets of $\mathbb N$. We use the fact that $J(k)$ has a finitely bounded homogeneous Ramsey expansion and that $G$ is a maximal closed subgroup of $\text{Sym}(V)$.

math.LO↗

arXiv2024

The filter of interpretability types of Hobby-McKenzie varieties is prime

We study the Hobby-McKenzie varieties that constitute a major class investigated thoroughly in the monograph The shape of congruence lattices by Kearnes and Kiss. We obtain new characterizations of the Hobby-McKenzie varieties via compatible reflexive ternary structures. Based on our findings, we prove that in the lattice of interpretability types of varieties, the filter of the interpretability types of Hobby-McKenzie varieties is prime.

math.CO↗

arXiv2023

Taylor is prime

We study the Taylor varieties and obtain new characterizations of them via compatible reflexive digraphs. Based on our findings, we prove that in the lattice of interpretability types of varieties, the filter of the types of all Taylor varieties is prime.

math.CO↗

arXiv2023

Symmetries of structures that fail to interpret something finite

We investigate structural implications arising from the condition that a given directed graph does not interpret, in the sense of primitive positive interpretation with parameters or orbits, every finite structure. Our results generalize several theorems from the literature and yield further algebraic invariance properties that must be satisfied in every such graph. Algebraic properties of this kind are tightly connected to the tractability of constraint satisfaction problems, and we obtain new such properties even for infinite countably categorical graphs. We balance these positive results by showing the existence of a countably categorical hypergraph that fails to interpret some finite structure, while still lacking some of the most essential algebraic invariance properties known to hold for finite structures.

cs.LO↗

arXiv2021

Permutation groups on countable vector spaces over prime fields

We describe all closed permutation groups which act on the set of vectors of a countable vector space $V$ over a prime field of odd order and which contain all automorphisms of $V$. In particular, we prove that their number is finite. These groups correspond, up to first-order interdefinability, precisely to all structures with a first-order definition in $V$.

math.LO↗

arXiv2021

HS-stability and complex products in involution semigroups

When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ${}^{*}$-semigroups and commutative involution semigroups.

math.RA↗

arXiv2021

Infinitely many reducts of homogeneous structures

It is shown that the countably infinite dimensional pointed vector space (the vector space equipped with a constant) over a finite field has infinitely many first order definable reducts. This implies that the countable homogeneous Boolean-algebra has infinitely many reducts. Our construction over the 2-element field is related to the Reed--Muller codes.

math.LO↗

arXiv2020

Classification of $ω$-categorical monadically stable structures

A first-order structure $\mathfrak{A}$ is called monadically stable iff every expansion of $\mathfrak{A}$ by unary predicates is stable. In this article we give a classification of the class $\mathcal{M}$ of $ω$-categorical monadically stable structures in terms of their automorphism groups. We prove in turn that $\mathcal{M}$ is smallest class of structures which contains the one-element pure set, closed under isomorphisms, and closed under taking finitely disjoint unions, infinite copies, and finite index first-order reducts. Using our classification we show that every structure in $\mathcal{M}$ is first-order interdefinable with a finitely bounded homogeneous structure. We also prove that every structure in $\mathcal{M}$ has finitely many reducts up to interdefinability, thereby confirming Thomas' conjecture for the class $\mathcal{M}$.

math.LO↗

arXiv2020

A Complexity Dichotomy in Spatial Reasoning via Ramsey Theory

Constraint satisfaction problems (CSPs) for first-order reducts of finitely bounded homogeneous structures form a large class of computational problems that might exhibit a complexity dichotomy, P versus NP-complete. A powerful method to obtain polynomial-time tractability results for such CSPs is a certain reduction to polynomial-time tractable finite-domain CSPs defined over k-types, for a sufficiently large k. We give sufficient conditions when this method can be applied and illustrate how to use the general results to prove a new complexity dichotomy for first-order expansions of the basic relations of the well-studied spatial reasoning formalism RCC5. We also classify which of these CSPs can be expressed in Datalog. Our method relies on Ramsey theory; we prove that RCC5 has a Ramsey order expansion.

math.LO↗

arXiv2020

Functional reducts of Boolean algebras

In this paper we classify some special reducts of the countable atomless Boolean algebra which we call functional reducts. We prove that there are exactly $13$ such structures up to first order interdefinability.

math.LO↗

arXiv2020

Structures with Small Orbit Growth

Let $K_{exp+}$ be the class of all structures $A$ such that the automorphism group of $A$ has at most $c n^{d n}$ orbits in its componentwise action on the set of $n$-tuples with pairwise distinct entries, for some constants $c,d$ with $d < 1$. We show that $K_{exp+}$ is precisely the class of finite covers of first-order reducts of unary structures, and also that $K_{exp+}$ is precisely the class of first-order reducts of finite covers of unary structures. It follows that the class of first-order reducts of finite covers of unary structures is closed under taking model companions and model-complete cores, which is an important property when studying the constraint satisfaction problem for structures from $K_{exp+}$. We also show that Thomas' conjecture holds for $K_{exp+}$: all structures in $K_{exp+}$ have finitely many first-order reducts up to first-order interdefinability.

math.GR↗

arXiv2015

Permutation groups containing infinite linear groups and reducts of infinite dimensional linear spaces over the two element field

Let $\mathbb{F}_2^ω$ denote the countably infinite dimensional vector space over the two element field and $\operatorname{GL}(ω, 2)$ its automorphism group. Moreover, let $\operatorname{Sym}(\mathbb{F}_2^ω)$ denote the symmetric group acting on the elements of $\mathbb{F}_2^ω$. It is shown that there are exactly four closed subgroups, $G$, such that $\operatorname{GL}(ω, 2)\leq G\leq \operatorname{Sym}(\mathbb{F}_2^ω)$. As $\mathbb{F}_2^ω$ is an $ω$-categorical (and homogeneous) structure, these groups correspond to the first order definable reducts of $\mathbb{F}_2^ω$. These reducts are also analyzed. In the last section the closed groups containing the infinite symplectic group $\operatorname{Sp}(ω, 2)$ are classified.

math.LO↗