Asymmetric Latin squares, Steiner triple systems, and edge-parallelisms

Peter J. Cameron

Published 2015-07-08Version 1

This article, showing that almost all objects in the title are asymmetric, is re-typed from a manuscript I wrote somewhere around 1980 (after the papers of Bang and Friedland on the permanent conjecture but before those of Egorychev and Falikman). I am not sure of the exact date. The manuscript had been lost, but surfaced among my papers recently. I am grateful to Laci Babai and Ian Wanless who have encouraged me to make this document public, and to Ian for spotting a couple of typos. In the section on Latin squares, Ian objects to my use of the term "cell"; this might be more reasonably called a "triple" (since it specifies a row, column and symbol), but I have decided to keep the terminology I originally used. The result for Latin squares is in B. D. McKay and I. M. Wanless, On the number of Latin squares, Annals of Combinatorics 9 (2005), 335-344 (arXiv 0909.2101), while the result for Steiner triple systems is in L. Babai, Almost all Steiner triple systems are asymmetric, Annals of Discrete Mathematics 7 (1980), 37-39.