Jan Hubička

Universal partial order represented by means of oriented trees and other simple graphs

We present several simple representations of universal partially ordered sets and use them for the proof of universality of the class of oriented trees ordered by the graph homomorphisms. This (which we believe to be a surprising result) solves several open problems. It implies for example universality of cubic planar graphs. This is in sharp contrast with representing even groups (and monoids) by automorphisms (and endomorphisms) of a bounded degree and planar graph. Thus universal partial orders (thin categories) are representable by much simpler structures than categories in general.

Finite Paths are Universal

Abstract We prove that any countable (finite or infinite) partially ordered set may be represented by finite oriented paths ordered by the existence of homomorphism between them. This (what we believe a surprising result) solves several open problems. Such path-representations were previously known only for finite and infinite partial orders of dimension 2. Path-representation implies the universality of other classes of graphs (such as connected cubic planar graphs). It also implies that finite partially ordered sets are on-line representable by paths and their homomorphisms. This leads to new on-line dimensions.

Finite presentation of homogeneous graphs, posets and Ramsey classes

It is commonly believed that one can prove Ramsey properties only for simple and “well behaved” structures. This is supported by the link of Ramsey classes of structures with homogeneous structures. We outline this correspondence in the context of the Classification Programme for Ramsey classes. As particular instances of this approach one can characterize all Ramsey classes of graphs, tournaments and partial ordered sets and also fully characterize all monotone Ramsey classes of relational systems (of any type). On the other side of this spectrum many homogeneous structures allow a concise description (called here a finite presentation) by means of all finite models of a suitable theory. Extending classical work of Rado (for the random graph) we find a finite presentation for each of the above classes where the classification problem is solved: (undirected) graphs, tournaments and partially ordered sets. The main result of the paper is a construction of classesP∈ andPf of finite structures which are isomorphic to the generic (i.e. homogeneous and universal) partially ordered set. Somehow surprisingly, the structureP∈ extends Conway’s surreal numbers and their linear ordering.

Bowtie-free graphs have a Ramsey lift

A bowtie is a graph consisting of two triangles with one vertex identified. We show that the class of all (countable) graphs not containing a bowtie as a subgraph has a Ramsey lift (expansion). This solves one of the old problems in the area and it is the first non-trivial Ramsey class with a non-trivial algebraic closure.

Universality of intervals of line graph order

We prove that for every d>=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.

Complexities of relational structures

Abstract The relational complexity, introduced by G. Cherlin, G. Martin, and D. Saracino, is a measure of ultrahomogeneity of a relational structure. It provides an information on minimal arity of additional invariant relations needed to turn given structure into an ultrahomogeneous one. The original motivation was group theory. This work focuses more on structures and provides an alternative approach. Our study is motivated by related concept of lift complexity studied by Hubička and Nešetřil.

Completing graphs to metric spaces

We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions.

An universality argument for graph homomorphisms

1 Preliminary results were already reported at Bordeaux graph theory workshop 2012 [2]. 2 Supported by MSMT CR grant LH12095 and GACR grant P202/12/G061. 3 Supported by grant ERC-CZ LL-1201 of the Czech Ministry of Education, CE-ITI of GACR P202/12/G061 and the European Associated Laboratory “Structures in Combinatorics” (LEA STRUCO) P202/12/6061. 4 Email: fiala@kam.mff.cuni.cz 5 Email: hubicka@iuuk.mff.cuni.cz 6 Email: longyangjing@gmail.com Available online at www.sciencedirect.com

Homomorphism-homogeneous L-colored graphs

A relational structure is homomorphism-homogeneous (HH-homogeneous for short) if every homomorphism between finite induced substructures of the structure can be extended to a homomorphism over the whole domain of the structure. Similarly, a structure is monomorphism-homogeneous (MH-homogeneous for short) if every monomorphism between finite induced substructures of the structure can be extended to a homomorphism over the whole domain of the structure. In this paper we consider L-colored graphs, that is, undirected graphs without loops where sets of colors selected from L are assigned to vertices and edges. A full classification of finite MH-homogeneous L-colored graphs where L is a chain is provided, and we show that the classes MH and HH coincide. When L is a diamond, that is, a set of pairwise incomparable elements enriched with a greatest and a least element, the situation turns out to be much more involved. We show that in the general case the classes MH and HH do not coincide.

Relations between graphs

Given two graphs G = ( V G ,  E G ) and H = ( V H ,  E H ) , we ask under which conditions there is a relation R ⊆  V G ×  V H that generates the edges of H given the structure of the graph G . This construction can be seen as a form of multihomomorphism. It generalizes surjective homomorphisms of graphs and naturally leads to notions of R-retractions, R-cores, and R-cocores of graphs. Both R-cores and R-cocores of graphs are unique up to isomorphism and can be computed in polynomial time.