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Jan Hubička

Publications and source records attributed to Jan Hubička.

46 records · Page 3Linked to original sources
arXiv2019

Automorphism groups and Ramsey properties of sparse graphs

We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todorčević correspondence. We investigate amenable and extremely amenable subgroups of these groups using the space of orientations of the graph and results from structural Ramsey theory. Resolving one of the open questions in the area, we show that Hrushovski's example of an $ω$-categorical sparse graph has no $ω$-categorical expansion with extremely amenable automorphism group.

math.GR↗
arXiv2017

Bowtie-free graphs have a Ramsey lift

A bowtie is a graph consisting of two triangles with one vertex identified. We show that the class of all (countable) graphs not containing a bowtie as a subgraph has a Ramsey lift (expansion). This solves one of the old problems in the area and it is the first non-trivial Ramsey class with a non-trivial algebraic closure.

math.CO↗
arXiv2017

Ramsey Classes with Closure Operations (Selected Combinatorial Applications)

We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Nešetřil-Rödl Theorem, the author's Ramsey lift of bowtie-free graphs as well as the Ramsey Theorem for Finite Models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of Structural Ramsey Theorem. We give here a more concise reformulation of recent authors paper "All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms)" and the main purpose of this paper is to show several applications. Particularly we prove the Ramsey property of ordered sets with equivalences on the power set, Ramsey theorem for Steiner systems, Ramsey theorem for resolvable designs and a partial Ramsey type results for $H$-factorizable graphs. All of these results are natural, easy to state, yet proofs involve most of the theory developed.

math.CO↗
arXiv2017

Fractal property of the graph homomorphism order

We show that every interval in the homomorphism order of finite undirected graphs is either universal or a gap. Together with density and universality this "fractal" property contributes to the spectacular properties of the homomorphism order. We first show the fractal property by using Sparse Incomparability Lemma and then by more involved elementary argument.

math.CO↗
arXiv2017

Ramsey theorem for designs

We prove that for any choice of parameters $k,t,λ$ the class of all finite ordered designs with parameters $k,t,λ$ is a Ramsey class.

math.CO↗
arXiv2014

Homomorphism and embedding universal structures for restricted classes

This paper unifies problems and results related to (embedding) universal and homomorphism universal structures. On the one side we give a new combinatorial proof of the existence of universal objects for homomorphism defined classes of structures (thus reproving a result of Cherlin, Shelah and Shi) and on the other side this leads to the new proof of the existence of dual objects (established by Nešetřil and Tardif). Our explicite approach has further applications to special structures such as variants of the rational Urysohn space. We also solve a related extremal problem which shows the optimality (of the used lifted arities) of our construction (and a related problem of A. Atserias).

math.CO↗
arXiv2014

Universality of intervals of line graph order

We prove that for every $d\geq 3$ the homomorphism order of the class of line graphs of finite graphs with maximal degree $d$ is universal. This means that every finite or countably infinite partially ordered set may be represented by line graphs of graphs with maximal degree $d$ ordered by the existence of a homomorphism.

math.CO↗
arXiv2013

Universal structures with forbidden homomorphisms

We relate the existence problem of universal objects to the properties of corresponding enriched categories (lifts or expansions). In particular, extending earlier results, we prove that for every (possibly infinite) regular set F of finite connected structures there exists a (countable) ω-categorical universal structure U for the class Forb(F) (of all countable structures not containing any homomorphic image of a member of F). We employ a technique known as homogenization: The universal object U is the shadow (reduct) of an ultrahomogeneous structure U'. We also put the results of this paper in the context of homomorphism dualities and constraint satisfaction problems. This leads to an alternative proof of the characterization of finite dualities (given by Tardif and Nešetřil) as well as of the characterization of infinite-finite dualities for classes of relational trees given by P. L. Erdős, Pálvölgyi, Tardif and Tardos. The notion of regular families of structures is motivated by the recent characterization of infinite-finite dualities for classes of relational forests (itself related to regular languages). We show how the notion of a regular family of relational trees can be extended to regular families of relational structures. This gives a partial characterization of the existence of a (countable) ω-categorical universal object for classes Forb(F).

math.CO↗