Jaroslav Nešetřil

Jaroslav Nešetřil
Charles University in Prague | CUNI · Computer Science Institute of Charles University

About

225
Publications
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5,529
Citations
Citations since 2016
55 Research Items
2090 Citations
20162017201820192020202120220100200300
20162017201820192020202120220100200300
20162017201820192020202120220100200300
20162017201820192020202120220100200300

Publications

Publications (225)
Preprint
Full-text available
Let $\mathscr C$ be a hereditary class of graphs. Assume that for every $p$ there is a hereditary NIP class $\mathscr D_p$ with the property that the vertex set of every graph $G\in\mathscr C$ can be partitioned into $N_p=N_p(G)$ parts in such a way that the union of any $p$ parts induce a subgraph in $\mathscr D_p$ and $\log N_p(G)\in o(\log |G|)$...
Preprint
Full-text available
Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures, and several important class properties can be defined in terms of transductions. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder...
Preprint
Full-text available
In this paper, we prove that for any $k\ge 3$, there exist infinitely many minimal asymmetric $k$-uniform hypergraphs. This is in a striking contrast to $k=2$, where it has been proved recently that there are exactly $18$ minimal asymmetric graphs. We also determine, for every $k\ge 1$, the minimum size of an asymmetric $k$-uniform hypergraph.
Chapter
In this paper, we prove that for any k≥3, there exist infinitely many minimal asymmetric k-uniform hypergraphs. This is in a striking contrast to k=2, where it has been proved recently that there are exactly 18 minimal asymmetric graphs. We also determine, for every k≥1, the minimum size of an asymmetric k-uniform hypergraph.
Chapter
We study Ramsey expansions of certain homogeneous 3-hypertournaments. We show that they exhibit an interesting behaviour and, in one case, they seem not to submit to current gold-standard methods for obtaining Ramsey expansions. This makes these examples very interesting from the point of view of structural Ramsey theory as there is a large demand...
Chapter
Using the Carlson–Simpson theorem, we give a new general condition for a structure in a finite binary relational language to have finite big Ramsey degrees.
Chapter
Algorithmic graph theory has been expanding at an extremely rapid rate since the middle of the twentieth century, in parallel with the growth of computer science and the accompanying utilization of computers, where efficient algorithms have been a prime goal. This book presents material on developments on graph algorithms and related concepts that...
Article
χ-bounded classes are studied here in the context of star colorings and, more generally, χp-colorings. This fits to a general scheme of sparsity and leads to natural extensions of the notion of bounded expansion class. In this paper we solve two conjectures related to star coloring (i.e. χ2) boundedness. One of the conjectures is disproved and in f...
Preprint
Full-text available
We study Ramsey expansions of certain homogeneous 3-hypertournaments. We show that they exhibit an interesting behaviour and, in one case, they seem not to submit to current gold-standard methods for obtaining Ramsey expansions. This makes these examples very interesting from the point of view of structural Ramsey theory as there is a large demand...
Preprint
Using the Carlson-Simpson theorem, we give a new general condition for a structure in a finite binary relational language to have finite big Ramsey degrees
Preprint
Full-text available
In this paper, we prove that for any $k\ge 3$, there exist infinitely many minimal asymmetric $k$-uniform hypergraphs. This is in a striking contrast to $k=2$, where it has been proved recently that there are exactly $18$ minimal asymmetric graphs. We also determine, for every $k\ge 1$, the minimum size of an asymmetric $k$-uniform hypergraph.
Article
In this brief paper we survey some of the recent development related to the notion of homomorphism. We briefly survey the many faceted development of the homomorphism concept in the last 50 years with emphasizes on recent developments.
Preprint
Full-text available
Inspired by a width invariant defined on permutations by Guillemot and Marx, the twin-width invariant has been recently introduced by Bonnet, Kim, Thomass\'e, and Watrigant. We prove that a class of binary relational structures (that is: edge-colored partially directed graphs) has bounded twin-width if and only if it is a first-order transduction o...
Article
We prove that if $G$ is a sparse graph — it belongs to a fixed class of bounded expansion $\mathcal{C}$ — and $d\in \mathbb{N}$ is fixed, then the $d$th power of $G$ can be partitioned into cliques so that contracting each of these clique to a single vertex again yields a sparse graph. This result has several graph-theoretic and algorithmic consequ...
Preprint
Full-text available
Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order transductions and the quasi-order they induce on infinite hereditary classes of finite graphs. Surprisingly, this quasi-order is very complex, though shaped by the locality properties of first-order...
Article
Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths – a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on...
Preprint
Full-text available
We study two notions of being well-structured for classes of graphs that are inspired by classic model theory. A class of graphs C is monadically stable if it is impossible to define arbitrarily long linear orders in vertex-colored graphs from C using a fixed first-order formula. Similarly, monadic dependence corresponds to the impossibility of def...
Article
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes...
Preprint
Full-text available
Szemerédi's Regularity Lemma is a very useful tool of extremal combinatorics. Recently, several refinements of this seminal result were obtained for special, more structured classes of graphs. We survey these results in their rich combinatorial context. In particular, we stress the link to the theory of (structural) sparsity, which leads to alterna...
Preprint
We prove that if $G$ is a sparse graph --- it belongs to a fixed class of bounded expansion $\mathcal{C}$ --- and $d\in \mathbb{N}$ is fixed, then the $d$th power of $G$ can be partitioned into cliques so that contracting each of these clique to a single vertex again yields a sparse graph. This result has several graph-theoretic and algorithmic con...
Article
We prove that the class of finite two-graphs has the extension property for partial automorphisms (EPPA, or Hrushovski property), thereby answering a question of Macpherson. In other words, we show that the class of graphs has the extension property for switching automorphisms. We present a short, self-contained, purely combinatorial proof which al...
Preprint
Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths. These results show a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views on c...
Preprint
Classes with bounded rankwidth are MSO-transductions of trees and classes with bounded linear rankwidth are MSO-transductions of paths -- a result that shows a strong link between the properties of these graph classes considered from the point of view of structural graph theory and from the point of view of finite model theory. We take both views o...
Article
Based on methods of structural convergence we provide a unifying view of local-global convergence, fitting to model theory and analysis. The general approach outlined here provides a possibility to extend the theory of local-global convergence to graphs with unbounded degrees. As an application, we extend previous results on continuous clustering o...
Preprint
Full-text available
We show the density theorem for the class of finite oriented trees ordered by the homomorphism order. We also show that every interval of oriented trees, in addition to be dense, is in fact universal. We end by considering the fractal property in the class of all finite digraphs.
Preprint
Let A \mathbf {A} be a finite structure. We say that a finite structure B \mathbf {B} is an extension property for partial automorphisms (EPPA)-witness for A \mathbf {A} if it contains A \mathbf {A} as a substructure and every isomorphism of substructures of A \mathbf {A} extends to an automorphism of B \mathbf {B} . Class C \mathcal C of finite st...
Chapter
We consider mappings, which are structure consisting of a single function (and possibly some number of unary relations) and address the problem of approximating a continuous mapping by a finite mapping. This problem is the inverse problem of the construction of a continuous limit for first-order convergent sequences of finite mappings. We solve the...
Preprint
We prove that the class of two-graphs has the extension property for partial automorphisms (EPPA, or Hrushovski property), thereby answering a question of Macpherson. In other words, we show that the class of graphs has the extension property for switching automorphisms. We present a short self-contained purely combinatorial proof which also proves...
Preprint
Full-text available
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes...
Preprint
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes...
Preprint
Full-text available
We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.
Conference Paper
Full-text available
The notion of bounded expansion captures uniform sparsity of graph classes and renders various algorithmic problems that are hard in general tractable. In particular, the model-checking problem for first-order logic is fixed-parameter tractable over such graph classes. With the aim of generalizing such results to dense graphs, we introduce classes...
Article
Full-text available
We prove that finite partial orders with a linear extension form a Ramsey class. Our proof is based on the fact that class of acyclic graphs has the Ramsey property and uses the partite construction.
Preprint
Full-text available
We consider mappings, which are structure consisting of a single function (and possibly some number of unary relations) and address the problem of approximating a continuous mapping by a finite mapping. This problem is the inverse problem of the construction of a continuous limit for first-order convergent sequences of finite mappings. We solve the...
Preprint
Based on methods of structural convergence we provide a unifying view of local-global convergence, fitting to model theory and analysis. The general approach outlined here provides a possibility to extend the theory of local-global convergence to graphs with unbounded degrees. As an application, we extend previous results on continuous clustering o...
Article
Cambridge Core - Recreational Mathematics - Connections in Discrete Mathematics - edited by Steve Butler
Chapter
We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Ne\v{s}et\v{r}il-R\"{o}dl 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 th...
Article
Full-text available
We study automorphism groups of sparse graphs from the viewpoint of topological dynamics and the Kechris, Pestov, Todor\v{c}evi\'c 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...
Article
Full-text available
We give Ramsey expansions of classes of generalised metric spaces where distances come from a linearly ordered commutative monoid. This complements results of Conant about the extension property for partial automorphisms and extends an earlier result of the first and the last author giving the Ramsey property of convexly ordered $S$-metric spaces....
Article
The recent increase of interest in the graph invariant called tree-depth and in its applications in algorithms and logic on graphs led to a natural question: is there an analogously useful "depth" notion also for dense graphs (say; one which is stable under graph complementation)? To this end, in a 2012 conference paper, a new notion of shrub-depth...
Article
We prove that for any choice of parameters $k,t,\lambda$ the class of all finite ordered designs with parameters $k,t,\lambda$ is a Ramsey class.
Article
Full-text available
We show that every free amalgamation class of finite structures with relations and (symmetric) partial functions is a Ramsey class when enriched by a free linear ordering of vertices. This is a common strengthening of the Ne\v{s}et\v{r}il-R\"odl Theorem and the second and third authors' Ramsey theorem for finite models (that is, structures with bot...
Article
Full-text available
We state the Ramsey property of classes of ordered structures with closures and given local properties. This generalises many old and new results: the Ne\v{s}et\v{r}il-R\"{o}dl 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 th...
Article
In this paper, we study a Ramsey type problems dealing with the number of ordered subgraphs present in an arbitrary ordering of a larger graph. Our first result implies that for every vertex ordered graph G on k vertices and any stochastic vector \(\overrightarrow{a}\) with k! entries, there exists a graph H with the following property: for any lin...
Book
This collection of high-quality articles in the field of combinatorics, geometry, algebraic topology and theoretical computer science is a tribute to Jiří Matoušek, who passed away prematurely in March 2015. It is a collaborative effort by his colleagues and friends, who have paid particular attention to clarity of exposition – something Jirka woul...
Article
We construct a Ramsey class whose objects are Steiner systems. In contrast to the situation with general $r$-uniform hypergraphs, it turns out that simply putting linear orders on their sets of vertices is not enough for this purpose: one also has to strengthen the notion of subobjects used from "induced subsystems" to something we call "strongly i...
Article
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 element...
Article
Full-text available
We prove the Ramsey property of classes of ordered structures with closures and given local properties. This generalises earlier results: the Ne\v{s}et\v{r}il-R\"odl Theorem, the Ramsey property of partial orders and metric spaces as well as the author's Ramsey lift of bowtie-free graphs. We use this framework to give new examples of Ramsey classes...
Article
The notion ofxxvi+803 structural sparsity is discussed, and its relation to the 'nowhere dense/somewhere dense' dichotomy introduced by the authors for classes of graphs is examined. The numerous facets of this dichotomy are surveyed, along with its connections to several concepts like stability, independence, VC-dimension, regularity partitions, e...
Article
We extend the general framework of structural limits from graphs and relational structures to finite structures (including function symbols). For perhaps the simplest model of this type - sets with single unary function - we determine limit objects with respect to the three main fragments of first order. In each of these cases we solve an analog of...
Article
Full-text available
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 have a Ramsey lift (expansion). This is the first non-trivial Ramsey class with a non-trivial algebraic closure.
Article
We exhibit explicit constructions of contractors for the graph parameter counting the number of B-flows of a graph, where B is a subset of a finite Abelian group closed under inverses. These constructions are of great interest because of their relevance to the family of B-flow conjectures formulated by Tutte, Fulkerson, Jaeger, and others.
Conference Paper
Full-text available
Recent characterization [9] of those graphs for which coloured MSO2 model checking is fast raised the interest in the graph invariant called tree-depth. Looking for a similar characterization for (coloured) MSO1, we introduce the notion of shrub-depth of a graph class. To prove that MSO1 model checking is fast for classes of bounded shrub-depth, we...
Chapter
Like every active area of mathematics our book leaves some intriguing and challenging problems open. They relate for example to better computations and improvements of provided bounds. In a way most of our results on logarithmic density are just the first order approximations which certainly can be improved on many places.
Chapter
An essential part of this book deals with estimates of complexity of algorithms. The aim of this chapter is to describe core algorithms, like the computation of a p-tree-depth decomposition. We shall describe this particular algorithm in a sufficiently precise way to allow an actual implementation of the described algorithms. In order to base our c...
Chapter
As a particular interpretation, this problem contains the decision problem for first-order logic (that is, the problem of algorithmically deciding whether a first-order formula is universally valid). A negative answer to the Entscheidungsproblem was given by Church and Turing, who proved that it is impossible to decide algorithmically whether state...
Chapter
Since the introduction by Alhazen and Avicenna of the experimental method and of the combination of observations, experiments and rational arguments in the early eleventh century, the scientific method gained in significance and became, after the works of Bacon, Descartes, Boyle and Newton, the standard methodology to come close to the Truth.
Chapter
3.1 For \(A \subseteq V(G)\) let f(A) be the number of edges between A and \( V(G)/A \). Let A be such that f(A) is maximal. No vertex \(\nu \in A\) has more neighbors in A than in V(G)\A as we would have f\((A-\nu) > A\).
Chapter
Meshes are a standard support for finding approximate solutions to partial differential equations (PDE) as well as of integral equations. Computational techniques working with a Divide-and-Conquer scheme need an initial mesh to be recursively broken into pieces of comparable size by cutting along small set of points (called a vertex separator).
Chapter
This chapter starts on an abstract level, by dealing with classes and their properties. As such it belongs to model theory (and general theory of categories). However our approach is very concrete and we deal with classes of graphs although many of the concepts and results carry over a more general setting. This will be made explicit in Sect. 5.8.
Chapter
After treating graph classes and class resolutions we return to the basics: the structure of finite trees as the true measure of our things.
Chapter
In the introduction we described the big picture of our theory. Here we begin with a more formal treatment. We define shallow minors, topological minors, and immersions as the basic local changes in graph classes. We show that edge densities in the iteration of these local changes are related and that they are also related to other parameters such...
Chapter
We know that the dichotomy nowhere dense vs. somewhere dense can be expressed by means of decompositions, maximal cliques, and coloring numbers. We now give in a way a dual characterization by independence(or stability).
Chapter
In the case where the input structures are restricted to some class C, some more “restricted” homomorphism dualities may appear.
Chapter
We start with the following observation: If a formula \( \Theta \)expresses a H-coloring problem (i.e.\( \textbf{G}\models \Theta\)if and only if\( G \rightarrow H\)the negated formula \( \rightharpoondown \Theta \)is preserved by homomorphisms: $$ \textbf{G}\models \rightharpoondown \Phi \;and\; G\rightarrow {G}\prime \quad \Rightarrow \quad {G}\p...
Chapter
The local properties of structures are frequently studied by means of decompositions: the large structure is cut into (hopefully simpler)pieces whose properties are then studied together with the interconnections between pieces. Several decomposition schemes can be considered. For example, one can stress the regularity of the interconnections of th...
Chapter
In this chapter we introduce the relevant concepts and techniques, and prove some basic results which will be used later on.
Chapter
An important part of the classical model theory studies properties of abstract mathematical structures (finite or not) expressible in first-order logic [257]. In the setting of finite model theory, which developed more recently, one studies first-order logic (and its various extensions) just on finite structures [141, 303]. Both theories share many...
Chapter
In this chapter we summarize the results on sparsity of classes with all their characterizations. The multiplicity of the equivalent characterizations that can be given for the nowhere dense–somewhere dense dichotomy is mainly a consequence of several related aspects:
Chapter
A matching of a graph G is a set of pairwise non-intersecting edges. An induced matching of a graph G is a matching of G which is an induced subgraph of G, that is a matching with the property that no endpoint of an edge in the matching is adjacent to an endpoint of another edge in the matching.
Chapter
Bounded expansion classes are the focus of this chapter and one of the leitmotivs of the whole book. In this chapter, we shall give many examples of classes with bounded expansion. The examples which we cover are schematically depicted on Fig. 14.1. These classes cover most classes considered in structural graph theory and the relevant parts of log...
Chapter
In the previous chapters of this book we investigated the problem of the existence of particular structures or operations with them, such as the existence of special partitions, the value of special parameters (for example \( \nabla_{r}(G) \) or the existence of special subsets.
Chapter
Combinatorics is a long story. But we believe that we live in the unique point of scientific history when combinatorics is becoming an essential part of mathematics and when the rich techniques developed in isolated and specific contexts are put to service in solving problems which are of general mathematical and scientific interest.
Article
We prove that the asymptotic logarithmic density of copies of a graph FF in the graphs of a nowhere dense class CC is integral and we determine the range of its possible values. This leads to a generalization of the trichotomy theorem of Nešetřil and Ossona de Mendez (2011) [18] and to a notion of the degree of freedom of a graph FF in a class CC....
Article
Each clone C on a fixed base set A induces a quasi-order on the set of all operations on A by the following rule: f is a C-minor of g if f can be obtained by substituting operations from C for the variables of g. By making use of a representation of Boolean functions by hypergraphs and hypergraph homomorphisms, it is shown that a clone C on {0,1} h...
Article
We answer a question raised by Lov\'asz and B. Szegedy [Contractors and connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a contractor for the graph parameter counting the number of B-flows of a graph, where B is a subset of a finite Abelian group closed under inverses. We prove our main result using the duality between flows an...
Article
We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories. KeywordsHomomorphisms-Structural theorems in combinatorics-Good characterization-Finite duality Mathematics Subject Classifications (2010)06D20-18B35-05C60
Article
This paper initiates a general study of the connection between graph homomorphisms and the Tutte polynomial. This connection can be extended to other polynomial invariants of graphs related to the Tutte polynomial such as the transition, the circuit partition, the boundary, and the coboundary polynomials. As an application, we describe in terms of...