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).
We prove the Ramsey property for classes of ordered structures with closures and given local properties. This generalises earlier results: the Nešetřil-Rödl 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 str…
Let C be a finite connected graph for which there is a countable universal C-free graph, and whose tree of blocks is a path. Then the blocks of C are complete. This generalizes a result of Furedi and Komjath, and fits naturally into a set of conjectures regarding the existence of countable C-free graphs, with C an arbitrary finite connected graph.
Partially ordered sets labeled with k labels (k-posets) and their homomorphisms are examined. We give a representation of directed graphs by k-posets; this provides a new proof of the universality of the homomorphism order of k-posets. This universal order is a distributive lattice. We investigate some other properties, namely the infinite distributivity, the computation of infinite suprema and infima, and the complexity of certain decision problems involving the homomorphism order of k-posets. Sublattices are also examined.
Colored Turán densities and finite directed templates
School of Mathematics, Shandong University（山东大学数学学院）；Universidad Pública de Navarra（纳瓦拉公立大学）；Extremal Combinatorics and Probability Group (ECOPRO), Institute for Basic Science (IBS)（基础科学研究院极端组合学与概率组）
We study countable embedding-universal and homomorphism-universal structures and unify results related to both of these notions. We show that many universal and ultrahomogeneous structures allow a concise description (called here a finite presentation). Extending classical work of Rado (for the random graph), we find a finite presentation for each of the following classes: homogeneous undirected graphs, homogeneous tournaments and homogeneous partially ordered sets. We also give a finite presentation of the rational Urysohn metric space and some homogeneous directed graphs. We survey well know…
We show that the problem of the existence of universal graphs with specified forbidden subgraphs can be systematically reduced to certain critical cases by a simple pruning technique which simplifies the underlying structure of the forbidden graphs, viewed as trees of blocks. As an application, we characterize the trees T for which a universal countable T-free graph exists.
In Chapter 1 we fully characterise pairs of finite graphs which form a gap in the full homomorphism order. This leads to a simple proof of the existence of generalised duality pairs. We also discuss how such results can be carried to relational structures with unary and binary relations. In Chapter 2 we show a very simple and versatile argument based on divisibility which immediately yields the universality of the homomorphism order of directed graphs and discuss three applications. In chapter 3, we show that every interval in the homomorphism order of finite undirected graphs is either univer…
Let F be a set of relational trees and let Forbh(F) be the class of all structures that admit no homomorphism from any tree in F; all this happens over a fixed finite relational signature $σ$. There is a natural way to expand Forbh(F) by unary relations to an amalgamation class. This expanded class, enhanced with a linear ordering, has the Ramsey property.
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 opti…
A $k$-edge-colored graph is a finite, simple graph with edges labeled by numbers $1,\ldots,k$. A function from the vertex set of one $k$-edge-colored graph to another is a homomorphism if the endpoints of any edge are mapped to two different vertices connected by an edge of the same color. Given a class $\mathcal{F}$ of graphs, a $k$-edge-colored graph $\mathbb{H}$ (not necessarily with the underlying graph in $\mathcal{F}$) is $k$-universal for $\mathcal{F}$ when any $k$-edge-colored graph with the underlying graph in $\mathcal{F}$ admits a homomorphism to $\mathbb{H}$. We characterize graph …
An $(m, n)$-colored-mixed graph $G=(V, A_1, A_2,\cdots, A_m, E_1, E_2,\cdots, E_n)$ is a graph having $m$ colors of arcs and $n$ colors of edges. We do not allow two arcs or edges to have the same endpoints. A homomorphism from an $(m,n)$-colored-mixed graph $G$ to another $(m, n)$-colored-mixed graph $H$ is a morphism $φ:V(G)\rightarrow V(H)$ such that each edge (resp. arc) of $G$ is mapped to an edge (resp. arc) of $H$ of the same color (and orientation). An $(m,n)$-colored-mixed graph $T$ is said to be $P_g^{(m, n)}$-universal if every graph in $P_g^{(m, n)}$ (the planar $(m, n)$-colored-mi…
We supply an upper bound on the distinguishing chromatic number of certain infinite graphs satisfying an adjacency property. Distinguishing proper $n$-colourings are generalized to the new notion of distinguishing homomorphisms. We prove that if a graph $G$ satisfies the connected existentially closed property and admits a homomorphism to $H$, then it admits continuum-many distinguishing homomorphisms from $G$ to $H$ join $K_2.$ Applications are given to a family universal $H$-colourable graphs, for $H$ a finite core.
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.
A partial order is universal if it contains every countable partial order as a suborder. In 2017, Fiala, Hubička, Long and Nešetřil 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, wh…
Using a categorial version of Fraïssé's theorem due to Droste and Göbel, we derive a criterion for a comma-category to have universal homogeneous objects. As a first application we give new existence result for universal structures and for ω-categorical universal structures. As a second application we characterize the retracts of a large class of homogeneous structures, extending previous results by Bonato, Delić, Dolinka, and Kubiś. As a third application we show for a large class of homogeneous structures that their polymorphism clone is generated by polymorphisms of bounded arity, generaliz…
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.
We survey structures endowed with natural partial orderings and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism order for various combinatorial objects.
We consider mappings between edge sets of graphs that lift tensions to tensions. Such mappings are called tension-continuous mappings (shortly TT mappings). Existence of a TT mapping induces a (quasi)order on the class of graphs, which seems to be an essential extension of the homomorphism order (studied extensively, see [Hell-Nesetril]). In this paper we study the relationship of the homomorphism and TT orders. We stress the similarities and the differences in both deterministic and random setting. Particularly, we prove that TT order is dense and universal and we solve a problem of M. DeVos …
In this paper we use the technique of Hopf algebras and quasi-symmetric functions to study the combinatorial polytopes. Consider the free abelian group $\mathcal{P}$ generated by all combinatorial polytopes. There are two natural bilinear operations on this group defined by a direct product $\times $ and a join $\divideontimes$ of polytopes. $(\mathcal{P},\times)$ is a commutative associative bigraded ring of polynomials, and $\mathcal{RP}=(\mathbb Z\varnothing\oplus\mathcal{P},\divideontimes)$ is a commutative associative threegraded ring of polynomials. The ring $\mathcal{RP}$ has the struct…
We present a construction that allows us to define a limit object of Banach space decorated graph sequences in a generalized homomorphism density sense. This general functional analytic framework provides a universal language for various combinatorial limit notions. In particular it makes it possible to assign limit objects to multigraph sequences that are convergent in the sense of node-and-edge homomorphism numbers, and it generalizes the limit theory for graph sequences with compact decorations.
