Peter Cameron's homepage

Welcome to my new homepage on GitHub. This page is under construction (and probably always will be!)

I am an Emeritus Professor in the School of Mathematics and Statistics at the University of St Andrews, and an Emeritus Professor of Mathematics at Queen Mary, University of London.

I am a Fellow of the Royal Society of Edinburgh.

About me My official page My diary zbMATH profile, MathSciNet profile, Google Scholar ORCID record: orcid.org/0000-0003-3130-9505 Mathematical genealogy, Wikipedia page CV and publications (PDF) Biographical stuff, travel diaries On this site Talks, publications, conjectures, coauthors, students Books: Introduction to Algebra, Combinatorics, Notes on Counting, Permutation Groups, Sets, Logic and Categories, ADE: Patterns in Mathematics, The Shrikhande graph (some dead links to be sorted out) Problems from my QMUL homepage

Elsewhere My homepage at QMUL Papers on arXiv Algebra and Combinatorics at St Andrews St Andrews pure mathematics colloquia CIRCA ORCiD Lecture notes Australasian Journal of Combinatorics Cameron Counts Mathematical quotes The Infinite Quest, a short course from the IAI Academy Mathematics: the next generation (LMS–Gresham lecture)

Research snapshots

Ramsey, Fraïssé, and orders

Among his discoveries was the fact that the structures in a nontrivial Ramsey class form a Fraïssé class (and so have a countable homogeneous Fraïssé limit), and are rigid (they have trivial automorphism group).

This was subsequently explained by the celebrated theorem of Kechris, Pestov and Todorčević, which uses topological dynamics (specifically, the automorphism group of the Fraïssé limit is extremely amenable) to show that this group has a total order as a reduct, and hence the structures in the class are totally ordered (and hence rigid).

Early in this story, I observed that there exist Fraïssé classes of rigid structures which have no "natural" total order, and wondered whether they could be Ramsey classes, and (after the KPT theorem showed that they were not) whether one could find explicit failures of the Ramsey property.

Siavash Lashkarighouchani and I have just done this. The paper is on the arXiv, at 2512.05684.

Old research snapshots are kept here.

I am Honorary Editor-In-Chief of the Australasian Journal of Combinatorics, an international open-access journal published by the Combinatorial Mathematics Society of Australasia.

A problem

Consider the following three graphs defined on a finite group G:

• The power graph: x and y are joined if one is a power of the other.
• The enhanced power graph: x and y are joined if they are each powers of some element w.
• The intersection power graph: x and y are joined if some non-identity element z is a power of each of them (and, by convention, the identity is joined to all other elements).

It is known that the power graph and enhanced power graph are equal if and only if every element of G has prime power order. After preliminary work by Higman and Suzuki, such groups were all determined by Brandl.

Problem: For which groups are the power graph and the intersection power graph equal?

It is known that groups in which all elements have prime order have this property, and that it implies that the power graph is a cograph. But an exact characterisation is unknown.

What we know about this question is in my paper with Sudip Bera, arXiv 2509.03919.

Old poblems are kept here.