University of St Andrews University of St Andrews

MT5861: Advanced Combinatorics

That's it!
That was the last presentation of the module
in its current form. Thank you all for your
attendance and comments, and best wishes
for whatever you do next.

The next presentation is scheduled for 2021-22, Semester 2 (commencing January 2022).

There are three syllabuses for this module, available here. In 2020-21, the third syllabus, on topics such as strongly regular graphs, root systems, designs, finite geometries, and Hadamard matrices, will be covered.

Course material from previous presentations of the module can be found here.

Assessed coursework

There will be two pieces of assessed coursework, each worth 10% of the marks for the module. The examination will be worth 80%.

The first assessed coursework is here, and the solutions are here. Marks and marked scripts are on MMS.

The second assessed coursework is here, and the solutions are here. Marks and marked scripts are on MMS.


The lectures are given in-person; video recordings will be available as technology permits.

Lecture, problems, and solutions

This table now includes short summaries of the lectures which may help those who cannot attend in person.

The sections in the table below correspond roughly to weeks, but with some slippage.

Added 12 March: Lecture 15 was not recorded due to a problem with Windows. Please see the summary below for the content of this lecture.

1Strongly regular graphsPart 1 Sheet 1, solutionsLect1, Lect2, Lect3
2Designs and projective planesPart 2Sheet 2, solutionsLect4, Lect5, Lect6
3Root systemsPart 3Sheet 3, solutionsLect7, Lect8, Lect9
4Graphs with least eigenvalue −2Part 4Sheet 4, solutions Lect10, Lect11
5Hadamard matricesPart 5Sheet 5, solutionsLect12, Lect13
6Projective spacesPart 6Revision1, solutionsLect14, Lect15
7Quadratic formsPart 7Sheet 7, solutions Lect16, Lect17
8The triangle propertyPart 8Sheet 8, solutions Lect18, Lect19
9Partitions into strongly regular graphsPart 9Revision 2, solutionsLect20, Lect21, Lect22

Supplementary material

I may post extra material about interesting topics related to the module, or PhD studentships that come to my attention, here.




Coursework sheets can be downloadad from the table above. You are welcome to hand in written work to be marked. I suggest on each sheet a maximum number of questions you should do. My solutions will be posted after the hand-in date.

Revision problems on the first half of the module are available here, and on the second half here; solutions are here and here.

Exam resources

Past exam papers on Syllabus 3 are available here. Note that the module was previously called MT5821: it was a 20-credit module and the exam was 2 hours 30 minutes.


I will make lecture notes available in advance of the lectures, so as to give you maximum flexibility in planning your time. But as usual I strongly advise you to attend the lectures, and then make use of the notes or slides to clarify things that were not clear from the lectures. Warning: The notes or slides may change, subject to correction of mistakes and comments from the audience!

Assessment will be 20% coursework and 80% examination.There will be a coursework sheet provided every week. These are optional except for two which will be clearly indicated in advance, which will be each worth 10% of the overall mark for the module.

Please do not hesitate to contact me by email if you have any questions.

Peter J. Cameron
17 April 2022