Once again there is not a huge amount to say in this post. Since the last post there have been a few additions to the polymath1 wiki that may be of some use. In particular, there is now a collection of fairly complete write-ups of related results (see the section entitled “Complete proofs or detailed sketches of potentially useful results”) to which I hope we will add soon. Also on the wiki is an account of the Ajtai-Szemerédi proof of the corners theorem, which seems to have some chance of serving as a better model for a proof of DHJ(3) than the proof via the triangle-removal lemma. Meanwhile, progress has been made in understanding and to some extent combinatorializing the ergodic-theoretic proof of DHJ(3), ideas from which have fed into the discussion. As with the last post, this one is mainly to stop the number of comments getting too large. We’re now down to 50 comments per post (except that it was 51 for the last one and will be 49 for this), since with the new threading we seem to be averaging at least one reply per comment.