# Robert WoodrowThe University of Calgary | HBI

Robert Woodrow

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56

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## Publications

Publications (56)

In his 2008 thesis, Tateno claimed a counterexample to the Bonato-Tardif conjecture regarding the number of equimorphy classes of trees. In this paper we revisit Tateno's unpublished ideas to provide a rigorous exposition, constructing locally finite trees having an arbitrary finite number of equimorphy classes; an adaptation provides partial order...

We study new partition properties of infinite Kn-free graphs. First, we investigate the number bpi(G,m) introduced by A. Aranda et al. (denoted there by r(G,m)) : the minimal r so that for any partition of G into r classes of equal size, there exists an independent set which meets at least m classes in size |G|. In the case of Henson’s countable un...

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

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...

A sibling of a relational structure $R$ is any structure $S$ which can be embedded into $R$ and, vice versa, in which $R$ can be embedded. Let $sib(R)$ be the number of siblings of $R$, these siblings being counted up to isomorphism. Thomass\'e conjectured that for countable relational structures made of at most countably many relations, $sib(R)$ i...

The lexicographic sum of graphs is defined as follows. Let be a graph. With each associate a graph . The lexicographic sum of the graphs over is obtained from by substituting each by . Given distinct , we have all the possible edges in the lexicographic sum between and if , and none otherwise. When all the graphs are isomorphic to some graph , the...

Our goal is to investigate a common relative of the independent transversal problem and the Dushnik–Erdős–Miller theorem in the class of infinite Kn-free graphs: we show that for any infinite Kn-free graph G=(V,E) and m∈N there is a minimal r=r(G,m) such that for any balanced r-colouring of the vertices of G one can find an independent set which me...

Two structures are said to be equimorphic if each embeds in the other. Such
structures cannot be expected to be isomorphic, and in this paper we
investigate the special case of linear orders, here also called chains. In
particular we provide structure results for chains having less than continuum
any isomorphism classes of equimorphic chains. We de...

Our goal is to investigate the independent transversal problem in the class of $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ so that for any balanced $r$-colouring of the vertices of $G$ one can find an independent set which meets at least $m$ colour classes in a set o...

Let V be a finite vertex set and let (𝔸, +) be a finite abelian group. An 𝔸-labeled and reversible 2-structure defined on V is a function g : (V × V) \ {(v, v) : v ∈ V } → 𝔸 such that for distinct u, v ∈ V, g(u, v) = −g(v, u). The set of 𝔸-labeled and reversible 2-structures defined on V is denoted by ℒ(V, 𝔸). Given g ∈ ℒ(V, 𝔸), a subset X of V is...

A 2-structure on a set $S$ is given by an equivalence relation on the set of
ordered pairs of distinct elements of $S$. A subset $C$ of $S$, any two
elements of which appear the same from the perspective of each element of the
complement of $C$, is called a clan. The number of elements that must be added
in order to obtain a 2-structure the only cl...

In 2005, Kechris, Pestov and Todorcevic provided a powerful tool to compute
an invariant of topological groups known as the universal minimal flow,
immediately leading to an explicit representation of this invariant in many
concrete cases. More recently, the framework was generalized allowing for
further applications, and the purpose of this paper...

It is assumed that a Kripke–Joyal semantics \({\mathcal{A} = \left\langle \mathbb{C},{\rm Cov}, {\it F},\Vdash \right\rangle}\) has been defined for a first-order language \({\mathcal{L}}\). To transform \({\mathbb{C}}\) into a Heyting algebra \({\overline{\mathbb{C}}}\) on which the forcing relation is preserved, a standard construction is used to...

We are interested in overgroups of the automorphism group of the Rado graph.
One class of such overgroups is completely understood; this is the class of
reducts. In this article we tie recent work on various other natural
overgroups, in particular establishing group connections between them and the
reducts.

Given a set S and a positive integer k, a binary structure is a function B:(S×S)∖{(x,x);x∈S}⟶{1,…,k}. The set S is denoted by V(B) and the integer k is denoted by rk(B). With each subset X of V(B) associate the binary substructure B[X] of B induced by X defined by B[X](x,y)=B(x,y) for any x≠y∈X. A subset X of V(B) is a clan of B if for any x,y∈X an...

The collection CL(T) of nonempty convex sublattices of a lattice T ordered by
bi-domination is a lattice. We say that T has the ?fixed point property for
convex sublattices (CLFPP for short) if every order preserving map f from T to
CL(T) has a fixed point, that is x > f(x) for some x > T. We examine which
lattices may have CLFPP. We introduce the...

A Sabidussi graph is defined from a total order T and a graph G as follows. Choose a vertex of G and denote it by 0. Denote by V(T)V(G) the family of the functions f : V(T)¡! V(G) such that fq2 V(T) : f(q)6= 0g is finite. The Sabidussi graph T G is defined on V(T)V(G) by: given f 6= g2(V(T)V(G)), f f; gg2 E(T G) if f f(m); g(m)g2 E(G), where m is t...

Considering an arbitrary relational structure on an infinite groundset, we analyze the implications of the following finiteness hypothesis (H): for some infinite cardinality μ there exist only finitely many isomorphism types of substructures of size μ. We show that the class C of relational structures satisfying (H) is intimately related to an expl...

Let T be a tournament whose arcs are coloured with k colours. Call a subset X of the vertices of Tabsorbing if from each vertex of T not in X there is a monochromatic directed path to some vertex in X. We consider the question of the minimum size of absorbing sets, extending known results and using new approaches. The greater part of the paper deal...

Abstract Following a question of Anstee and Farber we,investigate the possibility that all bridged graphs are cop-win. We show that in7nite chordal graphs, even of diameter two, need not be cop-win and point to some interesting questions, some of which we answer. c 2002 Elsevier Science B.V. All rights reserved. 1. Prologue

We describe several algorithms for finding the stable valuation of the vertices of an undirected graph from a partially defined valuation, subject to capacity constraints on the edges. We consider the complexities of the various algorithms and compare them in dependence to their uses, with emphasis on reusing the same network with its capacity cons...

Let G be a countable graph which has infinite chromatic number. Ifγis a coloring of [G]2with two colors, is there then a subsetH⊆Gsuch thatγis constant on [H]2andG|H,the graph induced by G onH,has infinite chromatic number? As edges and non-edges can be colored with different colors this will be the case iff G contains an infinite clique. It turns...

This volume contains the accounts of papers delivered at the Nato Advanced Study Institute on Finite and Infinite Combinatorics in Sets and Logic held at the Banff Centre, Alberta, Canada from April 21 to May 4, 1991. As the title suggests the meeting brought together workers interested in the interplay between finite and infinite combinatorics, se...

Let $L$ be a finite relational language. The age of a structure $\mathfrak{M}$ over $L$ is the set of isomorphism types of finite substructures of $\mathfrak{M}$. We classify those ages $\mathfrak{U}$ for which there are less than $2^\omega$ countably infinite pairwise nonisomorphic $L$-structures of age $\mathfrak{U}$.

LetE be a set and letL be a family of subsets ofE. A subsets ofE is called atransversal ofL ifs intersects each member ofL in exactly one element, that is |sl|=1, for everyl inL. We denoteT(L) the set of all transversals ofL. A pairB=(L, C) of families of subsets ofE is abox onE if it satisfies the following conditions:(i)
L=C=E, that is bothL andC...

Three results are obtained concerning the number of order preserving maps of an n-element partially ordered set to itself. We show that any such ordered set has at least 2
2n/3 order preserving maps (and 2
2
in the case of length one). Precise asymptotic estimates for the numbers of self-maps of crowns and fences are also obtained. In addition, low...

This paper is devoted to settling the following problem on (infinite, partially) ordered sets: Is there always a partition (2-coloring) of an ordered set X so that all nontrivial maximal chains of X meet both classes (receive both colors)? We show this is true for all countable ordered sets and provide counterexamples of cardinality ℵ 3 . Variants...

We consider closed permutation groups on a countably infinite set, and look at the permutation group induced on an infinite coinfinite subset. The automorphism group of the random graph has nice universality properties with respect to such subsets. Questions on when the stabiliser of the subset is faithful or closed on the subset are also considere...

We study infinite graphs in which every set of κ vertices has exactly λ common neighbours. We prove that there exist 2σ such graphs of each infinite order σ if κ is finite and that for κ infinite there are 2λ graphs of order λ and none of cardinality greater than λ (assuming the GCH). Further, we show that all a priori admissible chromatic numbers...

Afibre in a partially ordered set P is a subset of P meeting every maximal antichain of P. We give an example of a finite poset P with no one-element maximal antichain and containing no fibre of size at most |P}2, thus answering a question of Aigner-Andreae and disproving a conjecture of Lonc-Rival. We also proveTheorem 1.The elements of an arbitra...

For an integer k⩾2, a proper k-restraint on a graph G is a function from the vertex set of G to the set of k-colors. A graph G is amenably k-colorable if, for each nonconstant proper k-restraint r on G, there is a k-coloring c of G with c(v)≠r(v) for each vertex v of G. A graph G is amenable if it is amenably k-colorable and k is the chromatic numb...

What is the "archetypal" image that comes to mind when one thinks of an infinite graph? What with a finite graph - when it is thought of as opposed to an infinite one? What structural elements are typical for either - by their presence or absence - yet provide a common ground for both? In planning the workshop on "Cycles and Rays" it had been inten...

This problem is our starting point: Characterize those chains (linear orderings) C having a first and second element and with all principal final segments isomorphic to C. A description is obtained in the countable scattered case. For nonscattered countable chains, the problem is the same as the following: Characterize the colorings of the rational...

We prove that if a directed graph,D, contains no odd directed cycle and, for all but finitely many vertices, EITHER the in-degrees are finite OR the out-degrees are at most one, thenD contains an independent covering set (i.e. there is a kernel). We also give an example of a countable directed graph which has no directed cycle, each vertex has out-...

For any positive integer k let B(k) denote the bipartite graph of k- and k+1-element subsets of a 2k+1-element set with adjacency given by containment. It has been conjectured that for all k, B(k) is Hamiltonian. Any Hamiltonian cycle would be the union of two (perfect) matchings. Here it is shown that for all k>1 no Hamiltonian cycle in B(k) is th...

In this session we summarize the progress, as reported by participants of this meeting, on problems posed at the 1981 meeting in Banff. Because some presenters were reporting on others’ contributions, and since it seems impractical to repeat the background information published in the 1981 proceedings, we depart from the style of the other eight se...

J. PACH (presented by R.E. WOODROW): A graph G has property Ik if every k-element subset of vertices is contained in a finite isometric subgraph of G.

An ordered set (P,) has the m cutset property if for each x there is a set Fx with cardinality less than m, such that each element of Fx is incomparable to x and {x} Fx meets every maximal chain of (P,). Let n be least, such that each element x of any P having the m cutset property belongs to some maximal antichain of cardinality less than n. We sp...

An example is given of a complete theory with minimal models of arbitrarily large minimality rank.

Let G be a directed graph whose edges are coloured with two colours. Call a set S of vertices of Gindependent if no two vertices of S are connected by a monochromatic directed path. We prove that if G contains no monochromatic infinite outward path, then there is an independent set S of vertices of G such that, for every vertex x not in S, there is...

Let $G = \langle V_G, E_G \rangle$ be an undirected graph. The complementary graph G̃ is $\langle V_G, E_{\tilde{G}} \rangle$ where $(V_1, V_2) \in E_{\tilde{G}} \operatorname{iff} V_1 \neq V_1$ and $(V_1, V_2) \notin E_G$. Let K(n) be the complete undirected graph on n vertices and let E be the graph i.e. $\langle\{a, b, c\}, \{(b, c) (c, b)\}\ran...

Let (FORMULA PRESENTED), be an undirected graph. The complementary graph (FORMULA PRESENTED) where (FORMULA PRESENTED). Let K(n) be the complete undirected graph on n vertices and let E be the graph (FORMULA PRESENTED) i.e. (FORMULA PRESENTED) G is ultrahomogeneous just in case every isomorphism of subgraph of smaller cardinality can be lifted to a...

Thesis--Yale University. Includes bibliographical references (leaves 208-213). Photocopy of typescript. Ann Arbor, Mich. : University Microfilms International, 1980. -- 20 cm.

Let G be a countably infinite ultrahomogeneous undirected graph in which the complete graph on three vertices K3 cannot be embedded. Then G is isomorphic to one of the following four graphs: 1.(i) the countable graph on ω with no edges;2.(ii) the graph 〈ω, V〉 with V = {(2n, 2n + 1): nϵw} U{(2n + 1, 2n): nϵw}3.(iii) the graph 〈ω to, W〉 where W {(i,...

We give two examples. T
0 has nine countable models and a nonprincipal 1-type which contains infinitely many 2-types. T
1, has four models and an inessential extension T
2 having infinitely many models.

With quantifier elimination and restriction of language to a binary relation symbol and constant symbols it is shown that countable complete theories having three isomorphism types of countable models are "essentially" the Ehrenfeucht example [4, $\s6$].

Thesis (Ph. D.)--New Mexico State University, 1983. Includes vita. Includes bibliographical references (leaves 125-139). Microfilm. s

Typescript (photocopy). Thesis (Ph. D.)--New Mexico State University, 1983. Includes vita. Includes bibliographical references (leaves 125-139).

## Projects

Project (1)