On Thursday March 10, Peter Cholak gave a beautiful talk at the Logic Seminar at the University of Michigan, on Rado’s path decomposition theorem and its effective content. I want to review here some of the results covered by Peter. Slides for another version of the talk can be found in Peter’s page. This is joint work by Peter, Greg Igusa, Ludovic Patey, and Mariya Soskova.
As usual, given a set , let denote the collection of 2-sized subsets of . If is a positive integer, an –coloring of (or simply, a coloring, if is understood) is a map (where we use ordinal notation, so ). We can think of this as a coloring using colors of the edges of the complete graph whose underlying set of vertices is . When , we have an even simpler interpretation: A 2-coloring is just a graph on .
Given an -coloring of , a path of color is a sequence of distinct elements of (which may be finite or infinite, or even empty, or of length 1) such that for all , if is defined, then . Note that this is a much weaker requirement than asking that be monochromatic (which would mean that for all ). Also, in what follows is either a finite number or . However, we do not require that the elements in the sequence be listed in their natural order: We may very well have that for some .
The starting point is the following observation:
Fact (Erdős). If is finite and is a 2-coloring of , then there are paths of color 0 and of color 1 such that every (vertex) appears in exactly one of the
In general, if is an -coloring of , we say that , , is a path decomposition of iff each is a path of color and every vertex appears in exactly one of the . Using this notion, what the fact states is that for any finite , any 2-coloring of admits a path decomposition.
Proof. Suppose the result holds for and is a 2-coloring of . We can then find paths and of color 0 and 1 respectively such that each appears in exactly one of the . We want to show that the same holds for the full coloring (which includes edges one of whose vertices is ) at the possible expense of having to modify the partial paths we currently have. If one of the is empty, this is clear. Assume then that and . The result is also clear if or . Finally, if and , consider . If this color is 0, we can let the paths be and . Similarly, if , we can let the paths be and . (This is perhaps most obvious if a picture is drawn.)
Rado’s paper is a generalization of this result and its countable version. The reference is
MR0485504 (58 #5334)
Monochromatic paths in graphs.
Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977).
Ann. Discrete Math. 3 (1978), 191–194.
The paper opens indicating that Erdős sketched his proof to Rado; there does not seem to be an actual reference for Erdős’s proof. Rado proceeds to prove a more general version. I will only discuss here a particular case.
First, it should be noted that, unlike typical results in Ramsey theory where, once the case of two colors is handled, the argument easily generalizes to any number of colors, the proof above does not lift to more than two. The usual way of doing this lifting is by identifying all but one of the colors. This would result in two paths and , where along we only see color 0 and along we only see the other colors, but not 0. Let be the given coloring and be the set of vertices appearing in . If the restriction of to does not use color 0 we could indeed proceed inductively. But there is nothing to prevent 0 from being present as well, so the “easy” lifting argument actually breaks down.
The situation is indeed worse:
Theorem (Pokrovskiy). For any and any there is an and an -coloring of that does not admit a path decomposition.
The proof can be found in:
Partitioning edge-coloured complete graphs into monochromatic cycles and paths.
J. Combin. Theory Ser. B 106 (2014), 70–97.
On the other hand, we have:
Theorem (Rado). For any finite , any -coloring of admits a path decomposition.
As already mentioned, Rado’s result is more general, in particular allowing the use of countably many colors. However, the arguments that follow only apply directly to the stated version.
Before sketching the proof, note that even for , the result does not follow as usual from the finite version: Given a 2-coloring of , the standard approach would consist of letting be the paths resulting from successively applying Erdős’s theorem to the restrictions of to . But the inductive argument we presented allows the paths to be modified from one value of to the next, which means that we cannot ensure that the process will successfully identify (via initial segments) paths for the full coloring (the partial paths do not “stabilize”). Together with Pokrovskiy’s negative result just indicated, this leaves us with a curious Ramsey-theoretic statement to which the usual compactness arguments do not apply. (Its finite counterpart, Erdős’s result, is weaker in the sense that it only applies to two colors, and requires a different argument.)
Proof. Consider a nonprincipal ultrafilter on . The ultrafilter provides us with a notion of largeness. Given and a coloring , define for and the set of neighbors of in color as
Note that for any the partition and therefore there is exactly one such that is large (that is, it is in $\mathcal U$). For , define
and note that the partition .
We proceed by stages to define the paths as required. We set for all . In general, at the beginning of any given stage we have defined (finite) partial approximations to each path , say has length , with , using the convention that indicates that . For each , we will ensure that end extends (for all ), and simply set as the resulting path. Inductively, we require that each is a path of color , and that if , then .
Now, at stage , we simply consider the least not yet in any of the . There is a unique with . We set for all . If , then set . Finally, if , the point is that since and are both large, then so is their intersection (all we really need is that the intersection of sets in is nonempty). Let be a point in their intersection, and set . The induction hypothesis is preserved, and this completes stage of the construction.
It should be immediate that the so constructed indeed provide a path decomposition of , and this completes the proof.
It is interesting to note that the notation just developed allows us to give a quick proof of Ramsey’s theorem for pairs: Given a coloring , use notation as above, and note that for exactly one , the set is in . We argue that there is an infinite subset that is homogeneous for with color , that is, . Indeed, we can simply set , where the are defined recursively so that and for all .
As Peter indicated in his talk, these pretty arguments are somewhat dissatisfying in that invoking a nonprincipal ultrafilter is too strong a tool for the task at hand. He then proceeded to indicate how we can in fact do better, computationally speaking. For instance, if the coloring is computable, then we can find a path decomposition below . The key to this improvement comes from two observations.
First, we do not really need an ultrafilter to carry out the argument. It suffices to consider a set that is cohesive with respect to all the , meaning that is infinite and, for any , either or , where is the eventual containment relation: iff there is a finite subset of such that , in which case we say that is almost contained in .
The point is that we can replace all instances where we required that a set is in by the new largeness condition stating that is almost contained in . For instance, note that if are large, then so is their intersection. As before, for any there is a unique with large, and we can redefine as the set of such that
With these modifications, it is straightforward to verify that the proof above goes through. This shows that a path decomposition of is .
In more detail: Note first that these two conditions are indeed equivalent, and second, clearly the are pairwise disjoint since is infinite and, moreover, for all there is a unique such that :
Suppose that holds, and let be such that . Since is infinite, we can indeed find elements of larger than , and any such witnesses .
Conversely, if holds, then , because is cohesive and has infinite intersection with . But then holds, as wanted.
To see that any is in a unique , fix and use that is cohesive to conclude that if for all , then , which contradicts the infinitude of . It follows that for some and, since the are pairwise disjoint, this is unique. This proves that holds and therefore . Uniqueness follows from this same observation: If , then (as shown above) . But there is only one for which this is true.
The second observation is that there is an easy recursive construction of a set that is cohesive with respect to all the : Consider first . One of these sets is infinite (since their union is ), say , and let be its first element. Consider now
Their union is , so one of these sets is infinite, say . Let be its first element above . Etc. The set so constructed is as wanted. Note that this construction explicitly obtains an infinite set that, for each , is almost contained in one the , which is superficially stronger than being cohesive. However, as verified above, any set cohesive for all the must actually have this property.
Computationally, the advantage of this construction is that it makes explicit that all we need to access a cohesive set is an oracle deciding of any whether it is infinite. For computable , these are all questions.
Peter further refined this analysis in his talk via the notion of a set being over : This is any set such that for any uniformly computable sequence of pairs of sentences for such that at least one is true, there is an that predicts the true sentence of each pair in the sense that for all , if , then is true. In symbols, say that . The point of the notion is that a result of Jockusch and Stephen gives us that if then there is a cohesive set such that . The relevant paper is:
Jockusch, Carl; Stephan, Frank
A cohesive set which is not high.
Math. Logic Quart. 39 (1993), no. 4, 515–530.
Math. Logic Quart. 43 (1997), no. 4, 569.
This shows that a path decomposition for a computable coloring can actually be found below (and more).
Peter concluded his talk by indicating how for special colorings the complexity can be further improved. For instance, say that a coloring is stable iff exists for all . One can check that for stable , we can use cofinite as a notion of largeness in the preceding arguments, and that a path decomposition can accordingly be found when is computable below . On the other hand, this is optimal, in that one can find a stable computable such that any path decomposition computes .