Challenges in combinatorial design theory

Since WordPress changed their editor and it is no longer possible to write posts in HTML, I have to find a new solution to posting mathematics.

What I have done this time is to put it on my web page on GitHub (https://cameroncounts.github.io/web/) and link it from here.

On Thursday last week, Peter Keevash came to talk to the Pure Maths Colloquium in St Andrews about some of his really astonishing breakthroughs on the asymptotic existence and enumeration of combinatorial designs, including t-designs, Latin squares, Sudoku squares, and many others. But there are some problems he cannot solve, including Ryser’s and Brualdi’s conjectures on Latin squares. So I jotted down three problems as a challenge for magicians wielding the new powerful methods. The first, of course, is the existence of projective planes, which he mentioned in his talk. The second is a generalisation of the existence of resolvable designs; his methods may well solve this already. The third is a perhaps lesser known problem due to Mohammed Aljohani, John Bamberg and me, to which Steiner sytems are the proposed answer rather than the subject of the problem.