I’ve been in Paris for the weekend, so the number of comments on the previous post got rather large, and I also fell slightly behind. Writing this post will I hope help me catch up with what is going on.
FUNC with symmetry
One question that has arisen is whether FUNC holds if the ground set is the cyclic group and is rotationally invariant. This was prompted by Alec Edgington’s example showing that we cannot always find and an injection from to that maps each set to a superset. Tom Eccles suggested a heuristic argument that if is generated by all intervals of length , then it should satisfy FUNC. I agree that this is almost certainly true, but I think nobody has yet given a rigorous proof. I don’t think it should be too hard.
One can ask similar questions about ground sets with other symmetry groups.
A nice question that I came across on Mathoverflow is whether the intersection version of FUNC is true if consists of all subgroups of a finite group . The answers to the question came very close to solving it, with suggestions about how to finish things off, but the fact that the question was non-trivial was quite a surprise to me.