Talk:Outline of first paper
Discussion from the blog
(O'Donnell) 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.
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.
(Tao) I guess, on balance, that [k]={1,…,k} looks slightly nicer than [k]={0,…,k-1}; the 0-based notation is slightly more “logical”, but we don’t seem to derive any substantial benefit from it. So I’m weakly in favour of {1,…,k}.
We can borrow from geometry and use the notation Gr( [k]^n, d ) to denote the d-dimensional subspaces of [k]^n (a “combinatorial Grassmanian”). I don’t know what to call the measures on this space though. Does every measure \mu on [k]^n canonically define a measure on Gr( [k]^n, d )? It seems to me that one needs some additional parameters to specify such a measure.
“A” versus “{\mathcal A}” - I would prefer A, as I want to think of a subset of [k]^n as a set of points, rather than a collection of sets (which is what the {\mathcal A} notation suggests to me). The one problem with using A is that we are also likely to be using A for subsets of [n]^2 in the corners theorem, and if we are going to discuss the reduction of corners to DHJ then there might be a very slight notational clash there. But perhaps this is actually a good thing, since we want to use the corners argument to motivate the DHJ one…
As for the last question, what about “Hales-Jewett property” for containing lines, “subspace Hales-Jewett property” for containing subspaces, and “subspace Hales-Jewett-Varnavides property” for containing subspaces with positive probability? (thus, e.g. “dense ij-insensitive sets obey the subspace Hales-Jewett-Varnavides property”)?
