Homogeneous Steiner triple systems

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.