Thank you very much for the reference! For clarity in case anyone else comes across it, the Gumbel distribution is true only in the limit of dimension tending to infinity, but the authors do provide an explicit CDF for the finite dimensional case.

What a nice start! I'd personally been trying to use the Hadamard graphs as an example family with low independence number but was finding it hard to calculate the $|T|$. I agree with you regarding the dependence on $k$. I'll update the OP.

@PeterHeinig, thanks, I've taken your advice. I had used $d$ because it's the quantum dimension for the motivating problem, but obviously clarity here is more important.

@DavidE.Roberson pretty much spot on