A set of graphs is said to be independent if there is no homomorphism between distinct graphs from the set. We consider the existence problems related to the independent sets of countable graphs. While the maximal size of an independent set of countable graphs is 2^omega the On Line problem of extending an independent set to a larger independent set is much harder. We prove here that singletons can be extended (``partnership theorem''). While this is the best possible in general, we give structural conditions which guarantee independent extensions of larger independent sets. This is related to universal graphs, rigid graphs and to the density problem for countable graphs.
We establish links between countable algebraically closed graphs and the endomorphisms of the countable universal graph $R$. As a consequence we show that, for any countable graph $Γ$, there are uncountably many maximal subgroups of the endomorphism monoid of $R$ isomorphic to the automorphism group of $Γ$. Further structural information about End $R$ is established including that Aut $Γ$ arises in uncountably many ways as a Schützenberger group. Similar results are proved for the countable universal directed graph and the countable universal bipartite graph.
In this work we study the uncountable Borel chromatic numbers, defined by Geschke (2011) as cardinal characteristics of the continuum, of low complexity graphs. We show that a strong form of locally countable graphs with compact totally disconnected set of vertices have Borel chromatic number bounded by the continuum of the ground model in the model obtained by adding $\aleph_2$ Laver reals. From this, we answer a question from Geschke and the second author (2022), and another question from Fisher, Friedman and Khomskii (2014) concerning regularity properties of subsets of the real line.
For a countable ultrahomogeneous graph G let P(G) denote the collection of domains of subgraphs of G isomorphic to G. The order types of maximal chains in the set P(G) U ø ordered by the inclusion are characterized as: (I) the order types of compact sets of reals having the minimum non-isolated, if G is the Rado graph or the Henson graph H_n, for some n>2; (II) the order types of compact nowhere dense sets of reals having the minimum non-isolated, if G is the union of μdisjoint complete graphs of size ν, where μν=ω.
The distinguishing number of a graph $G$ is the smallest positive integer $r$ such that $G$ has a labeling of its vertices with $r$ labels for which there is no non-trivial automorphism of $G$ preserving these labels. Albertson and Collins computed the distinguishing number for various finite graphs, and Imrich, Klavžar and Trofimov computed the distinguishing number of some infinite graphs, showing in particular that the Random Graph has distinguishing number 2. We compute the distinguishing number of various other finite and countable homogeneous structures, including undirected and directed…
We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree at least three generating $μ$-hyperfinite equivalence relations admit $μ$-measurable matchings. We establish the analogous result for Baire measurable matchings in the locally finite case, and provide a c…
We give new examples and describe the complete lists of all measures on the set of countable homogeneous universal graphs and $K_s$-free homogeneous universal graphs (for $s\geq 3$) that are invariant with respect to the group of all permutations of the vertices. Such measures can be regarded as random graphs (respectively, random $K_s$-free graphs). The well-known example of Erdös--Rényi (ER) of the random graph corresponds to the Bernoulli measure on the set of adjacency matrices. For the case of the universal $K_s$-free graphs there were no previously known examples of the invariant measure…
Borel completeness of the class of countable Steiner triple systems
School of Mathematics\ University 24 South Section 1, First Ring Road\ , Sichuan 610065 China；School of Mathematics；Statistics\ Jiaotong University 3 Shangyuancun\ , Beijing 100044 China
In a 1977 paper, Steffens identified an elegant criterion for determining when a countable graph has a perfect matching. In this paper, we will investigate the proof-theoretic strength of this result and related theorems. We show that a number of natural variants of these theorems are equivalent, or closely related, to the ``big five'' subsystems of reverse mathematics. The results of this paper explore the relationship between graph theory and logic by showing the way in which specific changes to a single graph-theoretic principle impact the corresponding proof-theoretical strength. Taken tog…
We show that the existence of a universal structure implies the existence of a generic structure for any approximable class $\mathcal{C}$ of countable structures. We also show that the converse is not true. As a consequence, we provide several new examples of weak Fraïssé classes of finite graphs. Finally, we show that the class of all countable $C_4$-free graphs does not contain a generic structure, strengthening a result of A. Hajnal and J. Pach.
We prove that there is no countable universal $B_n$-free graph for all $n$ and that there is no countable universal graph in the class of graphs omitting all cycles of length at most $2k$ for $k\ge 2$.
The celebrated Erdős-Hajnal Conjecture says that in any proper hereditary class of finite graphs we are guaranteed to have a clique or anti-clique of size $n^c$, which is a much better bound than the logarithmic size that is provided by Ramsey's Theorem in general. On the other hand, in uncountable cardinalities, the model-theoretic property of stability guarantees a uniform set much larger than the bound provided by the Erdős-Rado Theorem in general. Even though the consequences of stability in the finite have been much studied in the literature, the countable setting seems a priori quite dif…
Given a countable graph, we say a set $A$ of its vertices is \emph{universal} if it contains every countable graph as an induced subgraph, and $A$ is \emph{weakly universal} if it contains every finite graph as an induced subgraph. We show that, for almost every graph on $\mathbb N$, $(1)$ every set of positive upper density is universal, and $(2)$ every set with divergent reciprocal sums is weakly universal. We show that the second result is sharp (i.e., a random graph on $\mathbb N$ will almost surely contain non-universal sets with divergent reciprocal sums) and, more generally, that neithe…
We construct a fully faithful functor from the category of graphs to the category of fields. Using this functor, we resolve a longstanding open problem in computable model theory, by showing that for every nontrivial countable structure S, there exists a countable field F with the same essential computable-model-theoretic properties as S. Along the way, we develop a new "computable category theory," and prove that our functor and its partially-defined inverse (restricted to the categories of countable graphs and countable fields) are computable functors.
Every countable graph can be built from finite graphs by a suitable infinite process, either adding new vertices randomly or imposing some rules on the new edges. On the other hand, a profinite topological graph is built as the inverse limit of finite graphs with graph epimorphisms. We propose to look at both constructions simultaneously. We consider countable graphs that can be built from finite ones by using both embeddings and projections, possibly adding a single vertex at each step. We show that the Rado graph can be built this way, while Henson's universal triangle-free graph cannot. We …
The random ordered graph is the up to isomorphism unique countable homogeneous linearly ordered graph that embeds all finite linearly ordered graphs. We determine the reducts of the random ordered graph up to first-order interdefinability.
We prove that in the Solovay model every OD graph G on reals satisfies one and only one of the following two conditions: (I) G admits an OD colouring by ordinals; (II) there exists a continuous homomorphism of G_0 into G, where G_0 is a certain F_sigma locally countable graph which is not R-OD colourable by ordinals in the Solovay model. If the graph G is locally countable or acyclic then (II) can be strengthened by the requirement that the homomorphism is a 1-1 map, i.e. an embedding. As the second main result we prove that Sigma^1_2 graphs admit the dichotomy (I) vs. (II) in set--generic ext…
We study semantic and syntactic properties of spherical orders and their elementary theories, including finite and dense orders and their theories. It is shown that theories of dense $n$-spherical orders are countably categorical and decidable. The values for spectra of countable models of unary expansions of $n$-spherical theories are described. The Vaught conjecture is confirmed for countable constant expansions of dense $n$-spherical theories.
In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel quasi-orders. In this paper we are concerned with universal countable Borel quasi-orders, i.e. countable Borel quasi-orders above all other countable Borel quasi-orders with regard to Borel reducibility. We…
We study the reverse mathematics and computability of countable graph theory, obtaining the following results. The principle that every countable graph has a connected component is equivalent to $\mathsf{ACA}_0$ over $\mathsf{RCA}_0$. The problem of decomposing a countable graph into connected components is strongly Weihrauch equivalent to the problem of finding a single component, and each is equivalent to its infinite parallelization. For graphs with finitely many connected components, the existence of a connected component is either provable in $\mathsf{RCA}_0$ or is equivalent to induction…
We investigate the poset (P(X),\subset), where P(X) is the set of isomorphic suborders of a countable ultrahomogeneous partial order X. For X different from (resp. equal to) a countable antichain the order types of maximal chains in (P(X)\cup \{\emptyset \},\subset) are characterized as the order types of compact (resp. compact and nowhere dense) sets of reals having the minimum non-isolated.
We show that the separative quotient of the poset (P(L),\subset) of isomorphic suborders of a countable scattered linear order L is σ-closed and atomless. So, under the CH, all these posets are forcing-equivalent (to P(ω)/Fin).
Metrically homogeneous graphs are connected graphs which, when endowed with the path metric, are homogeneous as metric spaces. Here we consider a class of countable metrically homogeneous graphs. The algebra of an age is a concept introduced by Cameron and is closely connected to the profile of the automorphism group of the associated countable structure. Cameron later provided sufficient structural conditions on the age of $\aleph_0$-categorical countable homogeneous structures for showing that the algebra of the age is a polynomial algebra. In this paper, we use Cameron's result to deduce th…
We show that the order dimension of the partial order of all finite subsets of $κ$ under set inclusion is ${\log}_{2}({\log}_{2}(κ))$ whenever $κ$ is an infinite cardinal. We also show that the order dimension of any locally countable partial ordering $(P, <)$ of size $κ^+$, for any $κ$ of uncountable cofinality, is at most $κ$. In particular, this implies that it is consistent with ZFC that the dimension of the Turing degrees under partial ordering can be strictly less than the continuum.
We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a proof-theoretic bound on definable countable ordinals whose existence is provable from the existence of second order indescribable cardinals over ZF.
The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary to prove theorems of ordinary mathematics, usually working in the language of second-order arithmetic $L_{2}$. A major theme in RM is therefore the study of structures that are countable or can be approximated by countable sets. Now, countable sets are represented by sequences here, because the usual higher-order definition of `countable set'cannot be expressed in $L_{2}$. Working in Kohlenbach's higher-order RM, we investigate various central theorems, e.g. those due to König, Ramsey, Bolzano, Weierstrass, a…
We investigate the statement "the order topology of every countable complete linear order is compact" in the framework of reverse mathematics, and we find that the statement's strength depends on the precise formulation of compactness. If we require that open covers must be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to $\mathsf{WKL}_0$ over $\mathsf{RCA}_0$. If open covers need not be uniformly expressible as unions of basic open sets, then the compactness of complete linear orders is equivalent to $\mathsf{ACA}_0$ over $\ma…
We prove that in some cases definable chains of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic chains, ROD chains in the Solovay model, and $Σ^1_2$ chains in the assumption that $ω_1^{L[x]}<ω_1$ for all reals $x$.
We prove that in some cases definable thin sets (including chains) of Borel partial orderings are necessarily countably cofinal. This includes the following cases: analytic thin sets, ROD thin sets in the Solovay model, and $Σ^1_2$ thin sets in the assumption that $ω_1^{L[x]}<ω_1$ for all reals $x$. We also prove that definable thin wellorderings admit partitions into definable chains in the Solovay model.
We show that a hypersimple unidimensional theory that has a club of reducts, in the partial order of all countable reducts, that are coordinatized in finite rank, is supersimple.
We describe all closed permutation groups which act on the set of vectors of a countable vector space $V$ over a prime field of odd order and which contain all automorphisms of $V$. In particular, we prove that their number is finite. These groups correspond, up to first-order interdefinability, precisely to all structures with a first-order definition in $V$.
The main theorem of this article is that every countable model of set theory M, including every well-founded model, is isomorphic to a submodel of its own constructible universe. In other words, there is an embedding $j:M\to L^M$ that is elementary for quantifier-free assertions. The proof uses universal digraph combinatorics, including an acyclic version of the countable random digraph, which I call the countable random Q-graded digraph, and higher analogues arising as uncountable Fraisse limits, leading to the hypnagogic digraph, a set-homogeneous, class-universal, surreal-numbers-graded acy…
It is well-known that the first order Peano axioms PA have a continuum of non-isomorphic countable models. The question, how close to being isomorphic such countable models can be, seems to be less investigated. A measure of closeness to isomorphism of countable models is the length of back-and-forth sequences that can be established between them. We show that for every countable ordinal alpha there are countable non-isomorphic models of PA with a back-and-forth sequence of length alpha between them. This implies that the Scott height (or rank) of such models is bigger than $α$. We also prove …
We indicate a way of distinguishing between structures, for which, two structures are said to be separable.Being separable implies being non-isomorphic. We show that for any first order theory $T$ in a countable language, if it has an uncountable set of countable models that are pairwise separable, then actually it has such a set of size $2^{\aleph_0}$. Our result follows trivially assuming the Continuum Hypothesis ($CH$). We work here in $ZFC$ (only without $CH$).
We consider the classification problem for several classes of countable structures which are "vertex-transitive", meaning that the automorphism group acts transitively on the elements. (This is sometimes called homogeneous.) We show that the classification of countable vertex-transitive digraphs and partial orders are Borel complete. We identify the complexity of the classification of countable vertex-transitive linear orders. Finally we show that the classification of vertex-transitive countable tournaments is properly above $E_0$ in complexity.
Given a countable scattered linear order $L$ of Hausdorff rank $α< ω_1$ we show that it has a $d\text{-}Σ_{2α+1}$ Scott sentence. Ash calculated the back and forth relations for all countable well-orders. From this result we obtain that this upper bound is tight, i.e., for every $α< ω_1$ there is a linear order whose optimal Scott sentence has this complexity. We further show that for all countable $α$ the class of Hausdorff rank $α$ linear orders is $\pmb Σ_{2α+2}$ complete.
We decide the Borel complexity of the conjugacy problem for automorphism groups of countable homogeneous digraphs. Many of the homogeneous digraphs, as well as several other homogeneous structures, have already been addressed in previous articles. In this article we complete the program, and establish a dichotomy theorem that this complexity is either the minimum or the maximum among relations which are classifiable by countable structures. We also discuss the possibility of extending our results beyond graphs to more general classes of countable homogeneous structures.
We consider strong combinatorial principles for sigma-directed families of countable sets in the ordering by inclusion modulo finite, e.g. P-ideals of countable sets. We try for principles as strong as possible while remaining compatible with CH, and we also consider principles compatible with the existence of nonspecial Aronszajn trees. The main thrust is towards abstract principles with game theoretic formulations. Some of these principles are purely combinatorial, while the ultimate principles are primarily combinatorial but also have aspects of forcing axioms.
