space of variable trees
I let a friend borrow my copy of the Discourses of Epictetus and now I miss it. It's good bedtime reading.
It's too bad that my waking hours are totally skewed. After work today I rode to MILO because I haven't seen anyone from there in a few days. But I forgot that most people really aren't awake at 1:45AM, despite my wanting them to be. In fact, the place was completely dead. Even my house, with all the apartments in it, have people up until 3 or 5AM, so I assume too much.
I had an idea today right before I got up from bed and I was still sleepy. Many Ramsey theorems can be proven as a consequence of some general partition theorem about variable words: the current 'most general' theorem keeps getting pushed further and further along. With it, most of the big results fall out in a way the literature describes fairly well.
My idea is to generalize the concept of variable word to variable tree. That is, associate letters and variables from some finite alphabet to the nodes of a finite (resp. infinite) tree, define some kind of concatenation and insertion rules. Then it becomes a semigroup so you can take its Stone-Cech compactification and use ultrafilter methods. Obviously a single branch is a tree, which corresponds exactly to a variable word, so we know we have all the previous results and then some. Let's hope it all goes through, I have to work on it more.
It's too bad that my waking hours are totally skewed. After work today I rode to MILO because I haven't seen anyone from there in a few days. But I forgot that most people really aren't awake at 1:45AM, despite my wanting them to be. In fact, the place was completely dead. Even my house, with all the apartments in it, have people up until 3 or 5AM, so I assume too much.
I had an idea today right before I got up from bed and I was still sleepy. Many Ramsey theorems can be proven as a consequence of some general partition theorem about variable words: the current 'most general' theorem keeps getting pushed further and further along. With it, most of the big results fall out in a way the literature describes fairly well.
My idea is to generalize the concept of variable word to variable tree. That is, associate letters and variables from some finite alphabet to the nodes of a finite (resp. infinite) tree, define some kind of concatenation and insertion rules. Then it becomes a semigroup so you can take its Stone-Cech compactification and use ultrafilter methods. Obviously a single branch is a tree, which corresponds exactly to a variable word, so we know we have all the previous results and then some. Let's hope it all goes through, I have to work on it more.
posted by Landon at 2:25 AM
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