Jaroslav Nešetřil

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• Professor (Full) at Charles University in Prague

Nešetřil and Ossona de Mendez recently proposed a new definition of graph convergence called structural convergence. The structural convergence framework is based on the probability of satisfaction of logical formulas from a fixed fragment of first-order formulas. The flexibility of choosing the fragment allows to unify the classical notions of con...

Structural convergence is a framework for the convergence of graphs by Nešetřil and Ossona de Mendez that unifies the dense (left) graph convergence and Benjamini-Schramm convergence. They posed a problem asking whether for a given sequence of graphs $(G_n)$ converging to a limit $L$ and a vertex $r$ of $L$, it is possible to find a sequence of ver...

For any integer $$h\geqslant 2$$ h ⩾ 2 , a set of integers $$B=\{b_i\}_{i\in I}$$ B = { b i } i ∈ I is a $$B_h$$ B h -set if all h -sums $$b_{i_1}+\ldots +b_{i_h}$$ b i 1 + … + b i h with $$i_1<\ldots <i_h$$ i 1 < … < i h are distinct. Answering a question of Alon and Erdős [2], for every $$h\geqslant 2$$ h ⩾ 2 we construct a set of integers X whic...

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been further extended to binary structures, in several (basically equivalent) ways. We prove that a class of binary rel...

This is an exposition of the contributions of László Lovász to mathematics and computer science written on the occasion of the bestowal of the Abel Prize 2021 to him. Our survey, of course, cannot be exhaustive. We sketch remarkable results that solved well-known open and important problems and that – in addition – had lasting impact on the develop...

Low treedepth decompositions are central to the structural characterizations of bounded expansion classes and nowhere dense classes, and the core of main algorithmic properties of these classes, including fixed-parameter (quasi) linear-time algorithms checking whether a fixed graph $F$ is an induced subgraph of the input graph $G$. These decomposit...

Inspired by a width invariant on permutations defined by Guillemot and Marx, Bonnet, Kim, Thomass\'e, and Watrigant introduced the twin-width of graphs, which is a parameter describing its structural complexity. This invariant has been further extended to binary structures, in several (basically equivalent) ways. We prove that a class of binary rel...

We give an infinitary extension of the Ne\v{s}et\v{r}il-R\"{o}dl theorem for category of relational structures with special type-respecting embeddings.

In this paper we define a new dimension of graphs based on the strong product. Strong product can be viewed as a categorical product in a modified category. Unlike in the standard case where the system of basic generators (“simplest objects”) is very transparent but necessarily infinite, we have here a single generator. Using just a single generato...

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 $\log N_p(G)\in o(\log |G|)$...

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...

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.

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.

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...

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.

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...

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...

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...

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

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.

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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...

We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces.

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...

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.

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...

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...

Cambridge Core - Recreational Mathematics - Connections in Discrete Mathematics - edited by Steve Butler

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...

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...

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...

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....

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...

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...

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.

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.

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

We prove the Ramsey property for 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 authors' Ramsey lift of bowtie-free graphs. We use this framework to solve several open problems and gi...

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...

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...

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...

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.

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.

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...

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.

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...

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...

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.

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\((A-\nu) > A\).

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).

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.

After treating graph classes and class resolutions we return to the basics: the structure of finite trees as the true measure of our things.

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...

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).

In the case where the input structures are restricted to some class C, some more “restricted” homomorphism dualities may appear.

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...

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...

In this chapter we introduce the relevant concepts and techniques, and prove some basic results which will be used later on.

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...