The following is a 2008 conjecture of Abraham, Bonnet and Kubiś: [ABK Conjecture] Every well quasi order (wqo) is a countable union of better quasi orders (bqo). We obtain a partial progress on the conjecture, by showing that the class of orders that are a countable union of better quasi orders (sigma-bqo) is closed under various operations. These include diverse products, such as the Dress-Shieffels product. We develop various properties of the latter product. In relation with the main question, we explore the class of alpha-wqo for countable ordinals alpha and obtain several closure properties and a Hausdorff-style classification theorem. Our main contribution is the discovery of various properties of sigma-bqos and ruling out potential counterexamples to the ABK Conjecture.

We present a structural approach of some results about jumps in the behavior of the profile (alias generating function) of hereditary classes of finite structures. We consider the following notion due to N.Thiéry and the second author. A monomorphic decomposition of a relational structure R is a partition of its domain V (R) into a family of sets (Vx)x∈X such that the restrictions of R to two finite subsets A and A′ of V (R) are isomorphic provided that the traces A∩Vx and A′ ∩Vx have the same size for each x ∈ X . Let Sμ be the class of relational structures of signature μ which do not have a finite monomorphic decomposition. We show that if a hereditary subclass D of Sμ is made of ordered relational structures then it contains a finite subset A such that every member of D embeds some member of A. Furthermore, for each R ∈ A the profile of the age A(R) of R (made of finite substructures of R) is at least exponential. We deduce that if the profile of a hereditary class of finite ordered structures is not bounded by a polynomial then it is at least exponential. This result is a part of classification obtained by Balogh, Bollobás and Morris (2006) for ordered graphs.

This paper is a contribution to the study of hereditary classes of relational structures, these classes being quasi-ordered by embeddability. It deals with the specific case of ordered sets of width two and the corresponding bichains and incomparability graphs. Several open problems about hereditary classes of relational structures which have been considered over the years have a positive answer in this case. For example, well-quasi-ordered hereditary classes of finite bipartite permutation graphs, respectively finite 321-avoiding permutations, have been characterized by Korpelainen, Lozin and Mayhill, respectively by Albert, Brignall, Ru & scaron;kuc and Vatter. In this paper we present an overview of properties of these hereditary classes in the framework of the Theory of Relations as presented by Roland Fra & iuml;ss & eacute;. We provide another proof of the results mentioned above. It is based on the existence of a countable universal poset of width two, obtained by the first author in 1978, his notion of multichainability (1978) (a kind of analog to letter-graphs), and metric properties of incomparability graphs. Using Laver's theorem (1971) on better-quasi-ordering (bqo) of countable chains we prove that a wqo hereditary class of finite or countable bipartite permutation graphs is necessarily bqo. This gives a positive answer to a conjecture of Nash-Williams (1965) in this case. We extend a previous result of Albert et al. by proving that if a hereditary class of finite, respectively countable, bipartite permutation graphs is wqo, respectively bqo, then the corresponding hereditary classes of posets of width at most two and bichains are wqo, respectively bqo. Several notions of labelled wqo are also considered. We prove that they are all equivalent in the case of bipartite permutation graphs, posets of width at most two and the corresponding bichains. We characterize hereditary classes of finite bipartite permutation graphs which remain wqo when labels from a wqo are added. Hereditary classes of posets of width two, bipartite permutation graphs and the corresponding bichains having finitely many bounds are also characterized. We prove that a hereditary class of finite bipartite permutation graphs is not wqo if and only if it embeds the poset of finite subsets of N ordered by set inclusion. This answers a long standing conjecture of the first author in the case of bipartite permutation graphs. (c) 2023 Elsevier Ltd. All rights reserved.

We show that every countable cograph has either one or infinitely many siblings. This answers, very partially, a conjecture of Thomasse. The main tools are the notion of well quasi-ordering and the correspondence between cographs and some labelled ordered trees.

We give several characteristic properties of FAC spaces, namely topological spaces with no infinite discrete subspace. The first one was obtained in 2019 by the first author, and states that every closed set is a finite union of irreducible closed subsets. The full result extends well-known characterizations of posets with no infinite antichain. One of them is that FAC spaces are, equivalently, topological spaces in which every closed set contains a dense Noetherian subspace, or spaces in which every Hausdorff subspace is finite, or in which no subspace has any infinite relatively Hausdorff subset. The latter comes with a nice min-max property, extending an observation of Erdős and Tarski in the case of posets: on spaces with no infinite relatively Hausdorff subset, the cardinalities of relatively Hausdorff subsets are bounded, and the least upper bound is also the least cardinality of a family of closed irreducible subsets that cover the space.

At the end of the forties, Fraïssé, following Cantor, Hausdorff and Sierpinski, highlighted the role of the embeddability quasi-order in the theory of relations. Since then, many results illustrating this role have been obtained (a large account was included in Fraïssé’s book Theory of Relations). In this paper, I present a selection of results centered on the notion of well-quasi-order (wqo). I mention several problems on wqo and hereditary classes of relational structures; some of these problems go back to the seventies.

We prove that if G=(V,E) is a nonprime graph with either no infinite independent set or no infinite clique, then every vertex of G belongs to a maximal strong module distinct from V. In particular, G admits a Gallai decomposition. As a consequence, we obtain that every interval order P with no infinite antichain admits a Gallai decomposition. That is, P is a lexicographical sum of interval orders distinct from P indexed by either a chain, an antichain, or a prime interval order. Next, we prove that every prime interval order with no infinite antichain is at most countable and does not embed a copy of the chain of rational numbers. Finally, for each countable ordinal α, we construct a well-quasi-ordered prime interval order P_α whose chain of maximal antichains has Hausdorff rank α.

The chain covering number $\Cov(P)$ of a poset $P$ is the least number of chains needed to cover $P$. For a cardinal $\nu$, we give a list of posets of cardinality and covering number $\nu$ such that for every poset $P$ with no infinite antichain, $\Cov(P)\geq \nu$ if and only if $P$ embeds a member of the list. This list has two elements if $\nu$ is a successor cardinal, namely $[\nu]^2$ and its dual, and four elements if $\nu$ is a limit cardinal with $\cf(\nu)$ weakly compact. For $\nu= \aleph_1$, a list was given by the first author; his construction was extended by F. Dorais to every infinite successor cardinal $\nu$.

We characterize the absolute retracts in the category of reflexive oriented graphs, that is, antisymmetric reflexive graphs, where morphisms between objects preserve arcs (which may be sent to loops). Here we show, by correcting a much earlier attempt at a proof, that a reflexive oriented graph is an absolute retract if and only if it is indeed a retract of some (direct) product of reflexive oriented zigzags (which are concatenations of reflexive oriented paths). Absolute retracts are therefore necessarily acyclic. In contrast to other categories of graphs and ordered sets, not every acyclic oriented graph can be embedded isometrically into some absolute retract. Embedding involves isometry with respect to the zig-zag distances forming a particular "dual quantale", which is a complete lattice of certain sets of words over the alphabet $\{+, -\}$, endowed with an additional monoid operation (viz., compound concatenation of sets of words) and an involution (interchanging $+$ and $- $ and then mirroring words). As reflexive oriented zigzags have MacNeille-closed distances, so do their products and retracts. So, the category of reflexive oriented graphs and its full subcategory of reflexive acyclic graphs do not have enough injectives, as the injective objects coincide with the absolute retracts.

We describe some metric properties of incomparability graphs. We consider the problem of the existence of infinite paths, either induced or isometric, in the incomparability graph of a poset. Among other things, we show that if the incomparability graph of a poset is connected and has infinite diameter, then it contains an infinite induced path. Furthermore, if the diameter of the set of vertices of degree at least 3 is infinite, then the graph contains as an induced subgraph either a comb or a kite.

The workshop "Homogeneous Structures, A Workshop in Honour of Norbert Sauer's 70th Birthday" took place at the Banff International Research Station from November 8th to November 13th, 2015. The purpose of the following note is to gather the list of open problems that were posed during the problem sessions. Contributions appear in alphabetical order, according to the name of their author.

We describe all well-quasi-ordered hereditary classes of countable or finite ordered sets coverable by two chains. This yields a similar description of the classes of associated bichains and incomparability graphs. In particular, we show that there are only countably many w.q.o. classes made of those finite structures. This includes previous results of Korpelainen, Lozin and Mayhill.

We present the Boolean dimension of a graph, we relate it with the notions of inner, geometric and symplectic dimensions, and with the rank and minrank of a graph. We obtain an exact formula for the Boolean dimension of a tree in terms of a certain star decomposition. We relate the Boolean dimension with the inversion index of a tournament.

We continue the investigation of systems of hereditarily rigid relations started in Couceiro, Haddad, Pouzet and Schölzel [1]. We observe that on a set $V$ with $m$ elements, there is a hereditarily rigid set $\mathcal{R}$ made of $n$ tournaments if and only if $m(m-1)\leq 2^{n}$ . We ask if the same inequality holds when the tournaments are replaced by linear orders. This problem has an equivalent formulation in terms of separation of linear orders. Let $h_{\text{Lin}}(m)$ be the least cardinal $n$ such that there is a family $\mathcal{R}$ of $n$ linear orders on an $m$ -element set $V$ such that any two distinct ordered pairs of distinct elements of $V$ are separated by some member of $\mathcal{R}$ , then $[\log_{2}(m(m-1))]\leq h_{\text{Lin}}(m)$ with equality if $m\leq 7$ . We ask whether the equality holds for every $m$ . We prove that $h_{\text{Lin}}(m+1)\leq h_{\text{Lin}}(m)+1$ . If $V$ is infinite, we show that $h_{\text{Lin}}(m)=\aleph_{0}$ for $m\leq 2^{\aleph_{0}}$ . More generally, we prove that the two equalities $h_{\text{Lin}}(m)=log_{2}(m)=d$ (Lin $(V)$ ) hold, where $\log_{2}(m)$ is the least cardinal $\mu$ such that $m\leq 2^{\mu}$ , and $d$ (Lin $(V)$ ) is the topological density of the set Lin (V) of linear orders on $V$ (viewed as a subset of the power set $\mathcal{P}(V\times V)$ equipped with the product topology). These equalities follow from the Generalized Continuum Hypothesis, but we do not know whether they hold without any set theoretical hypothesis.

The age $\mathcal{A}(G)$ of a graph $G$ (undirected and without loops) is the collection of finite induced subgraphs of $G$, considered up to isomorphy and ordered by embeddability. It is well-quasi-ordered (wqo) for this order if it contains no infinite antichain. A graph is \emph{path-minimal} if it contains finite induced paths of unbounded length and every induced subgraph $G'$ with this property embeds $G$. We construct $2^{\aleph_0}$ path-minimal graphs whose ages are pairwise incomparable with set inclusion and which are wqo. Our construction is based on uniformly recurrent sequences and lexicographical sums of labelled graphs.

A sibling of a relational structure $R$ is any structure $S$ which can be embedded into $R$ and, vice versa, such that $R$ can be embedded into $S$. Let $\operatorname{sib}(R)$ be the number of siblings of $R$, these siblings being counted up to isomorphism. Thomassé conjectured that for countable relational structures made of at most countably many relations, $\operatorname{sib}(R)$ is either one, countably infinite, or the size of the continuum; but even showing the special case $\operatorname{sib}(R)1$ is one or infinite is unsettled when $R$ is a countable tree. We prove that if $R$ is countable and $\aleph_{0}$-categorical, then indeed $\operatorname{sib}(R)$ is one or infinite. Furthermore, $\operatorname{sib}(R)$ is one if and only if $R$ is finitely partitionable in the sense of Hodkinson and Macpherson [14]. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in [35] and studied further in [23], [24], and a result of Frasnay [11].

Let $\mathrm{G}$ be a subgroup of the symmetric group $\mathfrak S(U)$ of all permutations of a countable set $U$. Let $\overline{\mathrm{G}}$ be the topological closure of $\mathrm{G}$ in the function topology on $U^U$. We initiate the study of the poset $\overline{\mathrm{G}}[U]:=\{f[U]\mid f\in \overline{\mathrm{G}}\}$ of images of the functions in $\overline{\mathrm{G}}$, being ordered under inclusion. This set $\overline{\mathrm{G}}[U]$ of subsets of the set $U$ will be called the \emph{poset of copies for} the group $\mathrm{G}$. A denomination being justified by the fact that for every subgroup $\mathrm{G}$ of the symmetric group $\mathfrak S(U)$ there exists a homogeneous relational structure $R$ on $U$ such that $\overline G$ is the set of embeddings of the homogeneous structure $R$ into itself and $\overline{\mathrm{G}}[U]$ is the set of copies of $R$ in $R$ and that the set of bijections $\overline G\cap \mathfrak S(U)$ of $U$ to $U$ forms the group of automorphisms of $\mathrm{R}$.

In this survey we present a generalization of the notion of metric space and some applications to discrete structures as graphs, ordered sets and transition systems. Results in that direction started in the middle eighties based on the impulse given by Quilliot (1983). Graphs and ordered sets were considered as kind of metric spaces, where - instead of real numbers - the values of the distance functions $d$ belong to an ordered semigroup equipped with an involution. In this frame, maps preserving graphs or posets are exactly the nonexpansive mappings (that is the maps $f$ such that $d(f(x),f(y))\leq d(x,y)$, for all $x,y$). It was observed that many known results on retractions and fixed point property for classical metric spaces (whose morphisms are the nonexpansive mappings) are also valid for these spaces. For example, the characterization of absolute retracts, by Aronszajn and Panitchpakdi (1956), the construction of the injective envelope by Isbell (1965) and the fixed point theorem of Sine and Soardi (1979) translate into the Banaschewski-Bruns theorem (1967), the MacNeille completion of a poset (1933) and the famous Tarski fixed point theorem (1955). This prompted an analysis of several classes of discrete structures from a metric point of view. In this paper, we report the results obtained over the years with a particular emphasis on the fixed point property.

We consider reflexive and involutive transition systems over an ordered alphabet A equipped with an involution. We give a description of the injective envelope of any two-element set in terms of Galois lattice, from which we derive a test of its finiteness. Our description leads to the notion of Ferrers language.