** — 1. Introduction — **

Recently, Florian Richter and I uploaded to the arXiv our paper titled `Large subsets of discrete hypersurfaces in contain arbitrarily many collinear points’.

This was the outcome of a fun project which started when we learned about Pomerance’s theorem:

Theorem 1 (Pomerance’s theorem)For every there exists such that for any with average gap bounded by , i.e.there exist points among which are collinear.

I have presented the proof of Pomerance’s theorem in this post. A weakening of Pomerance’s theorem was proved earlier by L. T. Ramsey (it is Lemma 1 in that paper) as a combinatorial intermediate step in a problem in Fourier analysis.

Lemma 2 (Ramsey’s lemma)For every there exists such that for any with gaps bounded by , i.e.there exist points among which are collinear.

Even when Lemma 2 is nontrivial (and in fact it is easy to derive Lemma 2 from the case when ).

The question we asked (and eventually answered) was whether these results could be extended to higher dimensions. The first problem here is how to even formulate such a generalization!

Continue reading