Comments on: What does it really mean to prove the Graph Minor Theorem?

By: Peter Gerdes

Peter Gerdes — Sat, 10 Sep 2011 14:47:55 +0000

To be clear the claim that for a graph with some particular fixed arithmetic graph minor relation (whether v1...vn is minor of w1...wm is an arithmetic property) is provable in ZF since there will always be an arithmetic graph exhibiting that minor relation. This is probably what Chow was saying, rather than my earlier interpretation which took him to just be requiring that you code up the graphs/minor relation using some nice primitive recursive system.

However, this is a bit of a cheat since that has the effect of bounding the total height of the minor relation to some fixed computable ordinal. As we know claims that go all the way up the computable ordinals can become crazily more complicated that any of the bounded versions, e.g., Kleene O is \Pi^1_1 complete while any initial segment is r.e. However, even though the claim that the minor theorem holds for all arithmetic graphs is not itself arithmetic Chow is right that this will still be provable from ZF.

For all my quibling ultimately the stronger closure conditions for constructable sets mean that the minor theorem restricted to those graphs that can be specified by use of the language of set theory is provable in ZF. So Chow is ultimately right that the TCS aspect can ignore choice.

By: Peter Gerdes

Peter Gerdes — Sat, 10 Sep 2011 13:37:04 +0000

The need (if indeed it is necessary) to use full choice is almost certainly a side effect of the application of the theorem to uncountable graphs. If you want the theorem for merely countable graphs you probably can get away with countable choice (which is entailed by both choice and determinacy).

Also note that definable sets all admit choice (if I is definable, the function going from i \in I to S_i is definable then there is a definable choice function for the collection S_i). This follows from the fact that L (the definable sets) is a model of ZFC.

Note that it is DEFINABILITY not the fixing of some means of coding vertices/minors by natural numbers that does the work. It's not enough to code up your infinite graphs using a nice coding function since some relations on vertexes are simply not possible to code arithmetically. Similar point applies for definable.

Of course any graph which can be specified in the language of arithmetic is fine.

Quick proof: Let G be the graph containing vertexes v_{-1},v_0...v_n... (countably infinite) and for i > 0 v_i has edges to v_0...v_{i-1}. Also v_i has an edge to v_{-1} iff i is in Kleene's O. Now any reasonable means of coding this graph as a set yields a representation for the graph that's non-arithmetic since otherwise we could arithmetically recover v_{-1} (only node with infinitely many edges) and each v_i and then check if i \in O which would make O arithmetic. Contradiction. Same reasoning to code in a graph which has no coded representation in L.

(sane here means the existence of edges between two nodes is arithmetic in the coded representation)

By: Tsuyoshi Ito

Tsuyoshi Ito — Fri, 02 Sep 2011 00:41:18 +0000

Aaron, thanks for the reply. I understand the intent. Maybe calling it “answer of the week” might be a little misleading. Something like “digging the treasure trove” (not sure if this is correct English) seems more accurate for this kind of posts. I should have said in my comment that I like both this post and the general idea of publicizing great questions and answers which were posted a while ago. Keep up the good work.

By: Aaron Sterling

Aaron Sterling — Thu, 01 Sep 2011 21:54:19 +0000

Thanks for the feedback, Tsuyoshi. I will be clearer about the dates in future posts. As a practical matter, it seems unavoidable that we will feature more "old" questions than "new" questions, because we feature at most one question a week, and the site generates more than one great answer per week. So we will never "catch up." Also, personally, I am interested in publicizing some of the great answers and discussions from the beta period of the site, which you know well, but most people following, eg, the TOC Aggregator, do not. Some of these posts will just be "refreshers" for the more dedicated users of the site.

By: Tsuyoshi Ito

Tsuyoshi Ito — Thu, 01 Sep 2011 21:09:08 +0000

That is undoubtedly a very good question and a very good answer. However, because this was posted as a blog (and also because it was filed under “Answer of the Week”), I expected that something new happened to the question, and I have to say that I was disappointed by the lack of new progress. There is nothing wrong with reviewing old questions and answers, but I would prefer if such reviews are clearly marked as such.