One paper in Euclidean Ramsey theory is “Transitive sets in Euclidean Ramsey theory” by Imre Leader, Paul A. Russell and Mark Walters. A set is transitive if its symmetry group of isometries is transitive. The paper looks at the question what if only transitive sets are Ramsey. It is known that Ramsey sets are spherical and it has been conjectured that all spherical sets are Ramsey. The paper shows that the conjectures are not equivalent that there exist sets that are spherical and cannot be embedded in any transitive set. It also looks at equivalent forms of the statement all transiive sets are Ramsey. It is here. I may talk further about the questions raised in this paper in future posts as there are some other papers written in response to these questions
I have just logged on. I have not posted in over a year. I think a while back I posted about new developments in the field. I might do that again. There are some things that I want to talk about in Euclidean Ramsey theory. I hope to get my next post here faster than the 18 month gap between this post and the last.
There is some news on this problem. There is a paper here.
It finds an example of size 1160 that has value 2 and proves that that example is maximal. The polymath 5 project found an example of size 1124. The proof that 1160 is maximal required a large computer proof. For more information see this post.
There is an uploaded version of 8a. Also there is going to be a article about the experience of working on 8a and possibly 8b for the newsletter of the European Mathematical society. The deadline is around April. For more check this post.
Polymath 9 has answered one of the questions it asked apparently. The idea that was being tried didn’t work there was a counterexample. It could possibly continue if the method could be modified to avoid the counterexample which is here. Meanwhile Polymath 8 has spawned a new project Polymath 8b. this is the latest post of this project.
Polymath 8 is working towards publication and it looks like Polymath 9 might be starting up. See this post.
There is a new Polymath project proposal. It is about bounded gaps between primes. The proposal is here.