Around homogeneity

TL;DR

This paper narrates the historical development and significance of homogeneous structures in permutation group theory, highlighting a specific graph and proposing a new generalization of Ramsey classes.

Contribution

It provides a historical account of a key graph in permutation groups and introduces the idea of Fra"{ss classes of rigid structures as a generalization of Ramsey classes.

Findings

Identification of a homogeneous triangle-free graph related to Higman--Sims graph

Historical insight into the development of permutation group theory

Proposal of Fra"{ss} classes of rigid structures as a new research direction

Abstract

Forty-five years ago, a young researcher in finite permutation group theory encountered a paper by Robert Woodrow. The homogeneous triangle-free graph Woodrow described there seemed to be an infinite analogue of the Higman--Sims graph which had played an important role in the researcher's thesis. The encounter changed the course of the researcher's career. This paper is the story of that event and its aftermath. The final section of the paper suggests that Fra\"{\i}ss\'e classes of rigid structures are a potentially interesting generalisation of Ramsey classes.