Homogeneous Steiner triple systems
Lead Research Organisation:
The Open University
Department Name: Mathematics & Statistics
Abstract
A combinatorial design is an arrangement of symbols into patterns in such a way that specified conditions are met. One example is a Sudoku, where each row of the square array contains each of the nine symbols exactly once, each column of the grid contains each of the nine symbols exactly once, and each mini-square contains each of the symbols exactly once. In a Sudoku the symbols are usually the numbers 1-9, but they could be any nine distinct symbols, colours, etc., since it is the arrangement that is important, not the actual symbols.
A Steiner triple system is another type of design; here a set of symbols is arranged into triples (sets of three symbols) in such a way that any pair of the symbols occurs together in exactly one triple.
In mathematics, homogeneity is a very strong symmetry property that means that any local symmetry is in fact a global symmetry. This project involves both finite and countably infinite Steiner triple systems and aims to work towards classifying all the Steiner triple systems with homogeneity properties.
A Steiner triple system is another type of design; here a set of symbols is arranged into triples (sets of three symbols) in such a way that any pair of the symbols occurs together in exactly one triple.
In mathematics, homogeneity is a very strong symmetry property that means that any local symmetry is in fact a global symmetry. This project involves both finite and countably infinite Steiner triple systems and aims to work towards classifying all the Steiner triple systems with homogeneity properties.
Planned Impact
This project concerns deep theoretical research, and so beyond the academic community, the impact of the research coming from this proposal is expected primarily to be indirect.
However, design theory is a field of combinatorics with close ties to several other areas of mathematics including group theory, finite fields, finite geometries, number theory, combinatorial matrix theory, and graph theory, and with a wide range of applications in areas such as information theory, statistics, computer science, biology, and engineering. The basic concepts of design theory are quite simple, but the mathematics used to study designs is varied, rich and ingenious. Advances in one area often have consequences in another. The underlying mathematical theory forms a solid foundation on which applied research can flourish.
Another example of impact stems from results on homogeneous structures. Homogeneous and omega-categorical structures have connections to constraint satisfaction problems and also to topological dynamics.
However, design theory is a field of combinatorics with close ties to several other areas of mathematics including group theory, finite fields, finite geometries, number theory, combinatorial matrix theory, and graph theory, and with a wide range of applications in areas such as information theory, statistics, computer science, biology, and engineering. The basic concepts of design theory are quite simple, but the mathematics used to study designs is varied, rich and ingenious. Advances in one area often have consequences in another. The underlying mathematical theory forms a solid foundation on which applied research can flourish.
Another example of impact stems from results on homogeneous structures. Homogeneous and omega-categorical structures have connections to constraint satisfaction problems and also to topological dynamics.
People |
ORCID iD |
| Bridget Webb (Principal Investigator) |
Publications
Bryant D
(2017)
On Hamilton decompositions of infinite circulant graphs
in Journal of Graph Theory
Bryant D
(2017)
On Hamilton Decompositions of Infinite Circulant Graphs
Horsley D
(2021)
Countable homogeneous Steiner triple systems avoiding specified subsystems
in Journal of Combinatorial Theory, Series A
| Description | The project comprised five aims: HC1, HF1-3 and CH1. Significant progress was made on HC1 with D. Horsley (Monash). We showed that a (finite) partial Steiner triple system has an embedding into a finite (complete) Steiner triple system which has no non-trivial proper subsystems that were not subsystems of the original partial Steiner triple system. This allows us to construct new countably infinite homogeneous Steiner triple systems. Paper in preparation. Significant progress was also made on HF1 and HF2, with D. Bryant and B. Maenhaut (UQ). We showed that Robinson's conjecture is true: that the Netto triple systems of prime order have no non-trivial subsystems. We have some general results on Netto triple systems, and in particular can describe the subsystems of the Netto triple systems of order p-cubed. Paper in preparation. Little progress was made on HF3: work started by considering both the Netto triple systems and the rank 7 triple systems, but to make progress effort was concentrated on the Netto triple systems. Progress was made on CH1 with D. Bryant, S. Herke and B. Maenhaut (UQ). We have further results that are leading towards resolving our conjecture on decomposing infinite circulant graphs into infinite Hamiltonian paths. Paper accepted and about to appear in J Graph Theory. |
| Exploitation Route | One paper is about to be published and two papers are in preparation, and some results have been presented (and well received) at international conferences (BCC Warwick 2015, ACCMCC UQ 2015, BCC Strathclyde 2017). Results presented at Invited Lectures (including at the University of South Wales) and at the Open University and will be presented at other international conferences. It is almost inevitable, when writing material up, that new results emerge, so the work is still ongoing. |
| Sectors | Digital/Communication/Information Technologies (including Software),Education,Other |
| URL | https://arxiv.org/abs/1701.08506 |