Claude Laflamme
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Claude Laflamme has a total of 18 co-authors. Here are the top co-authors by number of shared publications:
- Robert Woodrow
Department University of Calgary
12 shared publications - Maurice Pouzet
Department University of Calgary
9 shared publications - Norbert Sauer
Department University of Calgary
5 shared publications - Dwight Duffus
Emory University
3 shared publications - Andrés Aranda
Department of Applied Mathematics (KAM) Faculty of Mathematics and Physics Charles University Prague Czech Republic
2 shared publications - Dániel Tamás Soukup
Institute of Mathematics University of Vienna Kolingasse 14-16 1090 Wien, Austria
2 shared publications
Latest Publications
- 1 October 2023
- Published by Springer Nature in Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
- Vol. 93 (2) , 99-131
An example of Tateno disproving conjectures of Bonato–Tardif, Thomasse, and Tyomkyn
- 1 October 2023
- Published by Springer Nature in Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
- Vol. 93 (2) , 99-131
An example of Tateno disproving conjectures of Bonato–Tardif, Thomasse, and Tyomkyn
- Open Access
- 29 June 2021
- Published by University of Calgary in Contributions to Discrete Mathematics
- Vol. 16 (2) , 90-127
Siblings of an ℵ0-categorical relational structure
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].- 29 June 2021
- Published by University of Calgary in Contributions to Discrete Mathematics
- Vol. 16 (2) , 90-127
Open AccessSiblings of an ℵ0-categorical relational structure
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]. - Open Access
Distinguishing number of Urysohn metric spaces
The distinguishing number of a structure is the smallest size of a partition of its elements so that only the trivial automorphism of the structure preserves each cell of the partition. We show that for any countable subset of the positive real numbers, the corresponding countable homogeneous Urysohn metric space, when it exists, has distinguishing number 2 or the distinguishing number is infinite. While it is known that a sufficiently large finite primitive structure has distinguishing number 2, unless its automorphism group is the full symmetric group or alternating group, the infinite case is open and these countable Urysohn metric spaces provide further confirmation toward the conjecture that all primitive homogeneous countably infinite structures have distinguishing number 2 or else the distinguishing number is infinite.
Open AccessDistinguishing number of Urysohn metric spaces
The distinguishing number of a structure is the smallest size of a partition of its elements so that only the trivial automorphism of the structure preserves each cell of the partition. We show that for any countable subset of the positive real numbers, the corresponding countable homogeneous Urysohn metric space, when it exists, has distinguishing number 2 or the distinguishing number is infinite. While it is known that a sufficiently large finite primitive structure has distinguishing number 2, unless its automorphism group is the full symmetric group or alternating group, the infinite case is open and these countable Urysohn metric spaces provide further confirmation toward the conjecture that all primitive homogeneous countably infinite structures have distinguishing number 2 or else the distinguishing number is infinite.
- 30 September 2020
- Published by Elsevier in Discrete Mathematics
- Vol. 344 (1) , 112153
A universal partition result for infinite homogeneous -free and related graphs- 30 September 2020
- Published by Elsevier in Discrete Mathematics
- Vol. 344 (1) , 112153
A universal partition result for infinite homogeneous -free and related graphs- Open Access
The poset of copies for automorphism groups of countable relational structures
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 $\mathrm{R}$.Open AccessThe poset of copies for automorphism groups of countable relational structures
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 $\mathrm{R}$. - Open Access
Siblings of an $\aleph_0$-categorical relational structure
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, $sib(R)$ is either $1$, countably infinite, or the size of the continuum; but even showing the special case $sib(R)=1$ or infinite is unsettled when $R$ is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree $T$ the number of trees which are sibling of $T$ is either one or infinite. We prove that if $R$ is countable and $\aleph_{0}$-categorical, then indeed $sib(R)$ is one or infinite. Furthermore, $sib(R)$ is one if and only if $R$ is finitely partitionable in the sense of Hodkinson and Macpherson. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in a paper by Pouzet and Thi\'ery 2013 and studied further by Oudrar and Pouzet 2015, and a result of Frasnay 1984.Open AccessSiblings of an $\aleph_0$-categorical relational structure
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, $sib(R)$ is either $1$, countably infinite, or the size of the continuum; but even showing the special case $sib(R)=1$ or infinite is unsettled when $R$ is a countable tree. This is related to Bonato-Tardif conjecture asserting that for every tree $T$ the number of trees which are sibling of $T$ is either one or infinite. We prove that if $R$ is countable and $\aleph_{0}$-categorical, then indeed $sib(R)$ is one or infinite. Furthermore, $sib(R)$ is one if and only if $R$ is finitely partitionable in the sense of Hodkinson and Macpherson. The key tools in our proof are the notion of monomorphic decomposition of a relational structure introduced in a paper by Pouzet and Thi\'ery 2013 and studied further by Oudrar and Pouzet 2015, and a result of Frasnay 1984. - Open Access
Balanced independent sets in graphs omitting large cliques
Our goal is to investigate a close relative of the independent transversal problem in the class of infinite $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ such 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 of size $|V|$. Answering a conjecture of S. Thomass\'e, we express the exact value of $r(H_n,m)$ (using Ramsey-numbers for finite digraphs), where $H_n$ is Henson's countable universal homogeneous $K_n$-free graph. In turn, we deduce a new partition property of $H_n$ regarding balanced embeddings of bipartite graphs: for any finite bipartite $G$ with bipartition $A,B$, if the vertices of $H_n$ are partitioned into two infinite classes then there is an induced copy of $G$ in $H_n$ such that the images of $A$ and $B$ are contained in different classes.Open AccessBalanced independent sets in graphs omitting large cliques
Our goal is to investigate a close relative of the independent transversal problem in the class of infinite $K_n$-free graphs: we show that for any infinite $K_n$-free graph $G=(V,E)$ and $m\in \mathbb N$ there is a minimal $r=r(G,m)$ such 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 of size $|V|$. Answering a conjecture of S. Thomass\'e, we express the exact value of $r(H_n,m)$ (using Ramsey-numbers for finite digraphs), where $H_n$ is Henson's countable universal homogeneous $K_n$-free graph. In turn, we deduce a new partition property of $H_n$ regarding balanced embeddings of bipartite graphs: for any finite bipartite $G$ with bipartition $A,B$, if the vertices of $H_n$ are partitioned into two infinite classes then there is an induced copy of $G$ in $H_n$ such that the images of $A$ and $B$ are contained in different classes. - 10 May 2017
- Published by Springer Nature in Archive for Mathematical Logic
- Vol. 56 (7-8) , 811-829
Equimorphy: the case of chains
- 10 May 2017
- Published by Springer Nature in Archive for Mathematical Logic
- Vol. 56 (7-8) , 811-829
Equimorphy: the case of chains
- 4 January 2017
- Published by Springer Nature in Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
- Vol. 87 (2) , 369-408
Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif
- 4 January 2017
- Published by Springer Nature in Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg
- Vol. 87 (2) , 369-408
Invariant subsets of scattered trees and the tree alternative property of Bonato and Tardif
- Open Access
Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif
A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a function of the tree into the integers at least one of which is preserved by every embedding. With this result and a result of Tyomkyn, we prove that the tree alternative property conjecture of Bonato and Tardif holds for scattered trees and a conjecture of Tyomkin holds for locally finite scattered trees.
Open AccessInvariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif
A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a function of the tree into the integers at least one of which is preserved by every embedding. With this result and a result of Tyomkyn, we prove that the tree alternative property conjecture of Bonato and Tardif holds for scattered trees and a conjecture of Tyomkin holds for locally finite scattered trees.