Bridget S. Webb | Searcharxiv

arXiv2021

Countable homogeneous Steiner triple systems avoiding specified subsystems

In this article we construct uncountably many new homogeneous locally finite Steiner triple systems of countably infinite order as Fra\"ıssé limits of classes of finite Steiner triple systems avoiding certain subsystems. The construction relies on a new embedding result: any finite partial Steiner triple system has an embedding into a finite Steiner triple system that contains no nontrivial proper subsystems that are not subsystems of the original partial system. Fra\"ıssé's construction and its variants are rich sources of examples that are central to model-theoretic classification theory, and recently infinite Steiner systems obtained via Fra\"ıssé-type constructions have received attention from the model theory community.

math.CO↗

arXiv2015

Small Partial Latin Squares that Cannot be Embedded in a Cayley Table

We answer a question posed by Dénes and Keedwell that is equivalent to the following. For each order $n$ what is the smallest size of a partial latin square that cannot be embedded into the Cayley table of any group of order $n$? We also solve some variants of this question and in each case classify the smallest examples that cannot be embedded. We close with a question about embedding of diagonal partial latin squares in Cayley tables.

math.CO↗