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We prove a general Ramsey theorem for trees with a successor operation. This theorem is a common generalization of the Carlson-Simpson Theorem and the Milliken Tree Theorem for regularly branching trees. Our theorem has a number of applications both in finite and infinite combinatorics. For example, we give a short proof of the unrestricted Ne\v{s}et\v{r}il-R\"odl theorem, and we recover the Graham-Rothschild theorem. Our original motivation came from the study of big Ramsey degrees - various trees used in the study can be viewed as trees with a successor operation. To illustrate this, we give a non-forcing proof of a theorem of Zucker on big Ramsey degrees., Comment: 37 pages, 9 figures

Structural convergence is a framework for convergence of graphs by Ne\v{s}et\v{r}il 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 vertices $(r_n)$ such that $L$ rooted at $r$ is the limit of the graphs $G_n$ rooted at $r_n$. A counterexample was found by Christofides and Kr\'{a}l', but they showed that the statement holds for almost all vertices $r$ of $L$. We offer another perspective to the original problem by considering the size of definable sets to which the root $r$ belongs. We prove that if $r$ is an algebraic vertex (i.e. belongs to a finite definable set), the sequence of roots $(r_n)$ always exists.

We shall show that there exists only one duality pair for ordered graphs. We will also define a corresponding definition of $\chi$-boundedness for ordered graphs and show that all ordered graphs are $\chi$-bounded and prove an analogy of Gyarf\'as-Sumner conjecture for ordered graphs. We shall conclude with showing that these results hold for directed ordered graphs, but they do not hold for oriented ordered graphs.

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., Comment: 5 pages. Extended abstract

This is an exposition of the contributions of L\'aszl\'o Lov\'asz 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 development of subsequent research and even started whole new theories. Although discrete mathematics is what one can call the Lov\'asz home turf, his interests were, from the beginning of his academic career, much broader. He employed algebra, geometry, topology, analysis, stochastics, statistical physics, optimization, and complexity theory, to name a few, to contribute significantly to the explosive growth of combinatorics; but he also exported combinatorial techniques to many other fields, and thus built enduring bridges between several branches of mathematics and computer science. Topics such as computational convexity or topological combinatorics, for example, would not exist without his fundamental results. We also briefly mention his substantial influence on various developments in applied mathematics such as the optimization of real-world applications and cryptography., Comment: Chapter i Abel Prize book

Ne\v{s}et\v{r}il 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 convergence for sparse and dense graphs. Since the field is relatively young, the range of examples of convergent sequences is limited and only a few methods of construction are known. Our aim is to extend the variety of constructions by considering the gadget construction. We show that, when restricting to the set of sentences, the application of gadget construction on elementarily convergent sequences yields an elementarily convergent sequence. On the other hand, we show counterexamples witnessing that a generalization to the full first-order convergence is not possible without additional assumptions. We give several different sufficient conditions to ensure the full convergence. One of them states that the resulting sequence is first-order convergent if the replaced edges are dense in the original sequence of structures.

The notions of bounded-size and quasibounded-size decompositions with bounded treedepth base classes are central to the structural theory of graph sparsity introduced by two of the authors years ago, and provide a characterization of both classes with bounded expansions and nowhere dense classes. In this paper, we first prove that the model theoretic notions of dependence and stability are, for hereditary classes of graphs, compatible with quasibounded-size decompositions, in the following sense: every hereditary class with quasibounded-size decompositions with dependent (resp.\ stable) base classes is itself dependent (resp.\ stable). This result is obtained in a more general study of ``decomposition horizons'', which are class properties compatible with quasibounded-size decompositions. We deduce that hereditary classes with quasibounded-size decompositions with bounded shrubdepth base classes are stable. In the second part of the paper, we prove the converse. Thus, we characterize stable hereditary classes of graphs as those hereditary classes that admit quasibounded-size decompositions with bounded shrubdepth base classes. This result is obtained by proving that every hereditary stable class of graphs admits almost nowhere dense quasi-bush representations, thus answering positively a conjecture of Dreier et al. These results have several consequences. For example, we show that every graph $G$ in a stable, hereditary class of graphs $\mathscr C$ has a clique or a stable set of size $\Omega_{\mathscr C,\epsilon}(|G|^{1/2-\epsilon})$, for every $\epsilon>0$, which is tight in the sense that it cannot be improved to $\Omega_{\mathscr C}(|G|^{1/2})$.

We study various aspects of the first-order transduction quasi-order, which provides a way of measuring the relative complexity of classes of structures based on whether one can encode the other using a formula of first-order (FO) logic. In contrast with the conjectured simplicity of the transduction quasi-order for monadic second-order logic, the FO-transduction quasi-order is very complex; in particular, we prove that the quotient partial order is not a lattice, although it is a bounded distributive join-semilattice, as is the subposet of additive classes. Many standard properties from structural graph theory and model theory naturally appear in this quasi-order. For example, we characterize transductions of paths, cubic graphs, and cubic trees in terms of bandwidth, bounded degree, and treewidth. We establish that the classes of all graphs with pathwidth at most~$k$, for $k\geq 1$, form a strict hierarchy in the FO-transduction quasi-order and leave open whether same is true for treewidth. This leads to considering whether properties admit maximum or minimum classes in this quasi-order. We prove that many properties do not admit a maximum class, and that star forests are the minimum class that is not a transduction of a class with bounded degree, which can be seen as an instance of transduction duality. We close with a notion of dense analogues of sparse classes, and discuss several related conjectures. As a ubiquitous tool in our results, we prove a normal form for FO-transductions that manifests the locality of FO logic. This is among several other technical results about FO-transductions which we anticipate being broadly useful.

A partial order is universal if it contains every countable partial order as a suborder. In 2017, Fiala, Hubi\v{c}ka, Long and Ne\v{s}et\v{r}il showed that every interval in the homomorphism order of graphs is universal, with the only exception being the trivial gap $[K_1,K_2]$. We consider the homomorphism order restricted to the class of oriented paths and trees. We show that every interval between two oriented paths or oriented trees of height at least 4 is universal. The exceptional intervals coincide for oriented paths and trees and are contained in the class of oriented paths of height at most 3, which forms a chain., Comment: 6 pages, 1 figure. Extended abstract for Eurocomb 2021; corrected title

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., Comment: arXiv admin note: substantial text overlap with arXiv:2105.10031

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 for novel examples., Comment: Extended abstract accepted to EUROCOMB 2021

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, Comment: 6 pages, extended abstract accepted to EUROCOMB 2021

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., Comment: Accepted by EUROCOMB21

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 relational structures (that is: edge-colored partially directed graphs) has bounded twin-width if and only if it is a first-order transduction of a~proper permutation class. As a by-product, we show that every class with bounded twin-width contains at most $2^{O(n)}$ pairwise non-isomorphic $n$-vertex graphs.

Logical transductions provide a very useful tool to encode classes of structures inside other classes of structures. In this paper we study first-order (FO) transductions and the quasiorder they induce on infinite classes of finite graphs. Surprisingly, this quasiorder is very complex, though shaped by the locality properties of first-order logic. This contrasts with the conjectured simplicity of the monadic second order (MSO) transduction quasiorder. We first establish a local normal form for FO transductions, which is of independent interest. Then we prove that the quotient partial order is a bounded distributive join-semilattice, and that the subposet of \emph{additive} classes is also a bounded distributive join-semilattice. The FO transduction quasiorder has a great expressive power, and many well studied class properties can be defined using it. We apply these structural properties to prove, among other results, that FO transductions of the class of paths are exactly perturbations of classes with bounded bandwidth, that the local variants of monadic stability and monadic dependence are equivalent to their (standard) non-local versions, and that the classes with pathwidth at most $k$, for $k\geq 1$ form a strict hierarchy in the FO transduction quasiorder.

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 defining all graphs in this way. Examples of monadically stable graph classes are nowhere dense classes, which provide a robust theory of sparsity. Examples of monadically dependent classes are classes of bounded rankwidth (or equivalently, bounded cliquewidth), which can be seen as a dense analog of classes of bounded treewidth. Thus, monadic stability and monadic dependence extend classical structural notions for graphs by viewing them in a wider, model-theoretical context. We explore this emerging theory by proving the following: - A class of graphs $C$ is a first-order transduction of a class with bounded treewidth if and only if $C$ has bounded rankwidth and a stable edge relation (i.e. graphs from $C$ exclude some half-graph as a semi-induced subgraph). - If a class of graphs $C$ is monadically dependent and not monadically stable, then $C$ has in fact an unstable edge relation. As a consequence, we show that classes with bounded rankwidth excluding some half-graph as a semi-induced subgraph are linearly $\chi$-bounded. Our proofs are effective and lead to polynomial time algorithms.

Szemeredi'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 alternative proofs, refinements and solutions of open problems. It is interesting to note that many of these classes present challenging problems. Nevertheless, from the point of view of regularity lemma type statements, they appear as "gentle" classes.

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 consequences for powers of sparse graphs, including bounds on their subchromatic number and efficient approximation algorithms for the chromatic number and the clique number., Comment: 14 pages

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 classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes: 1) Graphs with linear rankwidth at most $r$ are linearly \mbox{$\chi$-bounded}. Actually, they have bounded $c$-chromatic number, meaning that they can be colored with $f(r)$ colors, each color inducing a cograph. 2) Based on a Ramsey-like argument, we prove for every proper hereditary family $\mathcal F$ of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in~$\mathcal F$. 3) For a class $\mathcal C$ with bounded linear rankwidth the following conditions are equivalent: a) $\mathcal C$~is~stable, b)~$\mathcal C$~excludes some half-graph as a semi-induced subgraph, c) $\mathcal C$ is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers., Comment: accepted at SODA 2020 conference. arXiv admin note: text overlap with arXiv:1909.01564

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 classes with bounded linear rankwidth and prove structural and model theoretic properties of these classes. The structural results we obtain are the following. 1) The number of unlabeled graphs of order $n$ with linear rank-width at most~$r$ is at most $\bigl[(r/2)!\,2^{\binom{r}{2}}3^{r+2}\bigr]^n$. 2) Graphs with linear rankwidth at most $r$ are linearly $\chi$-bounded. Actually, they have bounded $c$-chromatic number, meaning that they can be colored with $f(r)$ colors, each color inducing a cograph. 3) To the contrary, based on a Ramsey-like argument, we prove for every proper hereditary family $F$ of graphs (like cographs) that there is a class with bounded rankwidth that does not have the property that graphs in it can be colored by a bounded number of colors, each inducing a subgraph in $F$. From the model theoretical side we obtain the following results: 1) A direct short proof that graphs with linear rankwidth at most $r$ are first-order transductions of linear orders. This result could also be derived from Colcombet's theorem on first-order transduction of linear orders and the equivalence of linear rankwidth with linear cliquewidth. 2) For a class $C$ with bounded linear rankwidth the following conditions are equivalent: a) $C$ is stable, b) $C$ excludes some half-graph as a semi-induced subgraph, c) $C$ is a first-order transduction of a class with bounded pathwidth. These results open the perspective to study classes admitting low linear rankwidth covers.

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., Comment: 5 pages, 4 figures. Extended abstract for Eurocomb 2019. Minor revision

In this paper we prove a general theorem showing the extension property for partial automorphisms (EPPA, also called the Hrushovski property) for classes of structures containing relations and unary functions, optionally equipped with a permutation group of the language. The proof is elementary, combinatorial and fully self-contained. Our result is a common strengthening of the Herwig-Lascar theorem on EPPA for relational classes with forbidden homomorphisms, the Hodkinson-Otto theorem on EPPA for relational free amalgamation classes, its strengthening for unary functions by Evans, Hubi\v{c}ka and Ne\v{s}et\v{r}il and their coherent variants by Siniora and Solecki. We also prove an EPPA analogue of the main results of J. Hubi\v{c}ka and J. Ne\v{s}et\v{r}il: All those Ramsey classes (Ramsey classes with closures and forbidden homomorphisms), thereby establishing a common framework for proving EPPA and the Ramsey property. Our results have numerous applications, we include a solution of a problem related to a class constructed by the Hrushovski predimension construction., Comment: 63 pages, 3 figures. Minor revision addressing comments of the referee

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 also proves EPPA for the class of integer valued antipodal metric spaces of diameter 3, answering a question of Aranda et al. The class of two-graphs is an important new example which behaves differently from all the other known classes with EPPA: Two-graphs do not have the amalgamation property with automorphisms (APA), their Ramsey expansion has to add a graph, it is not known if they have coherent EPPA and even EPPA itself cannot be proved using the Herwig--Lascar theorem., Comment: 14 pages, 3 figures

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 of graphs with structurally bounded expansion, defined as first-order interpretations of classes of bounded expansion. As a first step towards their algorithmic treatment, we provide their characterization analogous to the characterization of classes of bounded expansion via low treedepth decompositions, replacing treedepth by its dense analogue called shrubdepth.

We present a short and self-contained proof of the extension property for partial isometries of the class of all finite metric spaces., Comment: 7 pages, 1 figure. Minor revision. Accepted to Commentationes Mathematicae Universitatis Carolinae

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 approximation problem and, consequently, the full characterization of limit objects for mappings for first-order (i.e. ${\rm FO}$) convergence and local (i.e. ${\rm FO}^{\rm local}$) convergence. This work can be seen both as a first step in the resolution of inverse problems (like Aldous-Lyons conjecture) and a strengthening of the classical decidability result for finite satisfiability in Rabin class (which consists of first-order logic with equality, one unary function, and an arbitrary number of monadic predicates). The proof involves model theory and analytic techniques.

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 of local convergent sequences and prove the existence of modeling quasi-limits for local-global convergent sequences of nowhere dense graphs., Comment: to appear in Commentationes Mathematicae Universitatis Carolinae

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 area, we show that Hrushovski's example of an $\omega$-categorical sparse graph has no $\omega$-categorical expansion with extremely amenable automorphism group., Comment: 41 pages, 2 figures, minor revision

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. Unlike Conant's approach, our analysis does not require the monoid to be semi-archimedean., Comment: 25 pages, 4 figures. Corrected proof of Lemma 6.20. Accepted to Contributions to Discrete Mathematics

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 has been introduced, such that it is related to the established notion of clique-width in a similar way as tree-depth is related to tree-width. Since then shrub-depth has been successfully used in several research papers. Here we provide an in-depth review of the definition and basic properties of shrub-depth, and we focus on its logical aspects which turned out to be most useful. In particular, we use shrub-depth to give a characterization of the lower ${\omega}$ levels of the MSO1 transduction hierarchy of simple graphs.

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., Comment: 8 pages, extended abstract for Eurocomb 2017

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 both relations and functions). We also find subclasses with the ordering property. For languages with relational symbols and unary functions we also show the extension property for partial automorphisms (EPPA) of free amalgamation classes. These general results solve several conjectures and provide an easy Ramseyness test for many classes of structures., Comment: 30 pages, 9 figures; corrections and presentation improvements suggested by the referee. Functions in structures are now set-valued

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 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., Comment: 16 pages, 2 figures. Minor correction according to referees comments. arXiv admin note: text overlap with arXiv:1606.07979. Author note: main theorem of arXiv:1606.07979 is used here

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 induced subsystems". Moreover we study the Ramsey properties of other classes of Steiner systems obtained from this class by either forgetting the order or by working with the usual notion of subsystems. This leads to a perhaps surprising induced Ramsey theorem in which designs get coloured., Comment: third version addresses changes arising from the referee reports

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., Comment: 8 pages, 5 figures. Minor corrections and reformatting. Accepted to European Journal of Combinatorics

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 give new examples of Ramsey classes. Among others, we find Ramsey lifts of convexly ordered $S$-metric spaces and prove the Ramsey theorem for finite models (i.e. structures with both functions and relations) thus providing the ultimate generalisation of the structural Ramsey theorem. Both of these results are natural, and easy to state, yet their proofs involve most of the theory developed here. We also characterise Ramsey lifts of classes of structures defined by finitely many forbidden homomorphisms and extend this to special cases of classes with closures. This has numerous applications. For example, we find Ramsey lifts of many Cherlin-Shelah-Shi classes., Comment: 91 pages, 21 figures. Accepted to Advances in Mathematics. Reformatted to match journal recommendations. Changed numbering of Theorems. Theorem 2.1 in the previous draft is now Theorem 2.11. Theorem 2.2 in the previous draft is now Theorem 2.18

The cluster analysis of very large objects is an important problem, which spans several theoretical as well as applied branches of mathematics and computer science. Here we suggest a novel approach: under assumption of local convergence of a sequence of finite structures we derive an asymptotic clustering. This is achieved by a blend of analytic and geometric techniques, and particularly by a new interpretation of the authors' representation theorem for limits of local convergent sequences, which serves as a guidance for the whole process. Our study may be seen as an effort to describe connectivity structure at the limit (without having a defined explicit limit structure) and to pull this connectivity structure back to the finite structures in the sequence in a continuous way., Comment: Patched version to allow compilation by arXiv