This is a continuation of the 1100-1199 thread of the polymath1 project, which is now full.  The focus has now mostly shifted to generalisations of the previous problems being studied to larger alphabet sizes k, so I am changing the title here from DHJ(3) to DHJ(k) to reflect this.

The discussion is evolving rapidly, but here are some of the topics currently being discussed:

• Understanding the density Hales-Jewett numbers $c_{n,k}$, defined as the size of the largest subset of $[k]^n$ that does not contain a combinatorial line.  The progress on that problem is summarised here.
• Fujimura’s problem in higher dimensions.
• The hyper-optimistic conjecture in higher dimensions.  It looks like the conjecture in fact fails in higher dimensions, though perhaps it could be salvaged by reformulating it.
• Reducing the need for computer assistance in our result $c'_5=124$ on Moser’s problem.
• New lower bounds for the coloring Hales-Jewett numbers.

Note that much of the most recent progress has not yet been ported to the wiki.  In order to help everyone else catch up, it may useful if authors of comments (particularly comments with lengthy computations, or with corrections to previous comments) put their work on the relevant page of the wiki (not necessarily in the most polished format), and perhaps only place a summary of it on the thread itself.

[Incidentally, for the more casual followers of this project, a non-technical introduction to this project can be found at this post of Jason Dyer.]

Comments here should start from 1200.