I haven’t posted here for a while. I have had as the title suggests had various problems. I have had a power outage then after I got that fixed My computer stopped working and then our internet went out. We ended up changing ISP’s which is good because the new one doesn’t have a data cap. Apparently the old one didn’t support our old modem. I got the computer working and I am posting this using the Linux operating system again. The problems I have had with this computer are related to hardware.
Happy Thanksgiving to everyone in the US. Recently I wrote about a problem I had with Linux. I think now that it is not related to Linux but was a hardware problem. In any case the problem has not repeated itself and I am glad.
I have recently had some trouble with Linux. Ubuntu wouldn’t load. I had to run fsck to fix it. I am not sure exactly what the problem was or what caused it. I may have similar problems. I will wait and see.
I have upgraded from Ubuntu 14.04 to 16.04. It made it easier to install polymake. So I installed it. It looks interesting. So far 16.04 is running smoothly for me.
I am using Linux as my main operating system for the first time. Before I had Chrome, before that Windows 7. It is OK. There are a few odd things like I have to watch Netflix on a Chrome browser and Hulu on Firefox but it is going well. I can run Sage on this computer without an emulator so in that sense it is an improvement over Windows. Also I don’t need antivirus software.
I have not posted for a while. One thing I think I mentioned a while back is computer problems. That has been dealt with to a degree that it should not be an obstacle to posting. I hope everyone is enjoying the holidays and that they will have a good 2016.
In my previous post “transitive sets” I discussed the idea that instead of all spherical sets being Ramsey, a set is Ramsey if its symmetry group of isometries is transitive. In this post I want to give a simple example of a configuration which is spherical but can’t be embedded into a set which has a transitive symmetry group of isometries.
The example comes from the paper “Transitive sets and cyclic quadrilaterals” by I. Leader,P.A. Russell and Mark Walters. It is here. In corollary 2 of the paper they prove the set $$((-1,0),(1,0),(a,\sqrt{1-a^2},(a,-\sqrt{1-a^2}))$$ where $a$ is transcendental does not embed into any transitive set. This type of set is called a kite.
If you could find a set which satisfies the above properties and prove it is Ramsey that would provide a counterexample to the hypothesis of the first paragraph. In any case this is a good example of a set which is spherical and can’t be embedded into a transitive set. I may continue this series of posts later.
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.
