Related: mathoverflow.net/questions/43317/…

@FedorPetrov: only the prime case of Olson's theorem follows from Alon-Furedi hyperplane covering. Though that has been "fixed" recently by Clark et al. (and by Brink before that) by generalising another result of Alon-Furedi. See section 4 of arxiv.org/pdf/1404.7793v2.pdf. In fact many combinatorial results that follow from combinatorial nullstellensatz can alternately be seen as direct applications of this Alon-Furedi bound.

@Louis: Thanks. That looks like a good candidate.

Thanks for the clarification and for the reference. My office mate works on this stuff and I think he would enjoy reading that article as well.

Are you sure that the example is due to Henry Cohn? From what I have heard in lectures on F1 theory, including one lecture by Manin, this example was due to Jacques Tits.

You are welcome. You can also find the proof of Larman, Rogers and Seidel here: mathoverflow.net/questions/17006/…