On the order of countable graphs

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

Constructing universal graphs for induced-hereditary graph properties

Rado constructed a (simple) denumerable graph R with the positive integers as vertex set with the following edges: For given m and n with m < n, m is adjacent to n if n has a 1 in the m'th position of its binary expansion. It is well known that R is a universal graph in the set Ic of all countable graphs (since every graph in Ic is isomorphic to an induced subgraph of R). In this paper we construct graphs which are universal in or for P for di erent inducedhereditary properties P of countable graphs. Constructions of universal graphs for the graph properties containing all graphs with colouring-number at most k+1 and k-degenerate graphs are obtained by restricting the edges of R. Results on the properties of these graphs are given and relationships between them are explored. This is followed by a general recursive construction which proves the existence of a countable universal graph for any induced-hereditary property of countable general graphs. A general construction of universal graphs for products of properties of graphs is also presented. The paper is concluded by a comparison between the two types of constructions of universal graphs.Research of the third author was supported by VEGA Grant No. 2/0194/10.http://link.springer.com/journal/12175hb201

Universality for and in induced-hereditary graph properties

The well-known Rado graph R is universal in the set of all countable graphs I, since every countable graph is an induced subgraph of R. We study universality in I and, using R, show the existence of 20 pairwise non-isomorphic graphs which are universal in I and denumerably many other universal graphs in I with prescribed attributes. Then we contrast universality for and universality in induced-hereditary properties of graphs and show that the overwhelming majority of induced-hereditary properties contain no universal graphs. This is made precise by showing that there are 2(20 ) properties in the lattice K< of induced-hereditary properties of which only at most 20 contain universal graphs. In a final section we discuss the outlook on future work; in particular the question of characterizing those induced-hereditary properties for which there is a universal graph in the property.http://www.discuss.wmie.uz.zgora.pl/gt/am201

Combinatorial Properties of Finite Models

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 known structures that are finitely presented. We focus on structures endowed with natural partial orders and prove their universality. These partial orders include partial orders on sets of words, partial orders formed by geometric objects, grammars, polynomials and homomorphism orders for various combinatorial objects. We give a new combinatorial proof of the existence of embedding-universal objects for homomorphism-defined classes of structures. This relates countable embedding-universal structures to homomorphism dualities (finite homomorphism-universal structures) and Urysohn metric spaces. Our explicit construction also allows us to show several properties of these structures.Comment: PhD thesis, unofficial version (missing apple font

The random graph

Erd\H{o}s and R\'{e}nyi showed the paradoxical result that there is a unique (and highly symmetric) countably infinite random graph. This graph, and its automorphism group, form the subject of the present survey.Comment: Revised chapter for new edition of book "The Mathematics of Paul Erd\H{o}s

Universal graphs with forbidden subgraphs and algebraic closure

We apply model theoretic methods to the problem of existence of countable universal graphs with finitely many forbidden connected subgraphs. We show that to a large extent the question reduces to one of local finiteness of an associated''algebraic closure'' operator. The main applications are new examples of universal graphs with forbidden subgraphs and simplified treatments of some previously known cases