Talk:Outline of first paper

Discussion from the blog

One question: It looks like you’ve divided the proof into three main lemmas: multidim-Sperner (more generally, multidim-DHJ(k-1)), line-free set correlating with intersections of ij-insensitive sets, and ij-insensitive sets being partitionable.

It seems to me that the Varnavides-version of multidim-Sperner (more generally, multidim-DHJ(k-1)) may as well be considered the basic lemma. Where will this go?

Putting it into \subsection{The multidimensional Sperner theorem} makes sense to me, although then the actual \section{A proof of the theorem for $k=3$.} might be quite short. On the other hand, if it goes into the proof section itself, then the multidim-Sperner therem subsection will be awfully short (might as well just quote Gunderson-Rodl-Sidorenko).

The latter seems less modular to me, so I guess what I’m ultimately suggesting is that \subsection{The multidimensional Sperner theorem} be more like \subsection{The Varnavides multidimensional Sperner theorem}.

Except that I strongly vote for using a more generic descriptor than “Varnavides”. There’s got to be a catchy word that indicates to the reader that not only do dense sets contain lines, a random line is in there with positive probability.

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Also, I’m still of two minds as to whether “Equal-Slices” should be treated as the main distribution, with Polya as a slight variant, or vice versa.