What is the best lower bound known to you? The best lower bound to my knowledge is linear ($2d$). I only know a construction with minimum degree at least $2^{\sqrt d}$ due to N. Alon. Do you have the references to the examples you mentioned?

@AndréHenriques I would like to hear some stories about shy mathematicians. :) Is that also a bad idea to point out the mistakes of others?

@Dirk more specific for mathematics, in the sense that I'm not talking about data, experiment and so on, I'm talking about proofs and statements.