Peter Krautzberger · on the web

Hindman's Theorem, partial semigroups and some of my most lacking intuitions (part 4)

Well, I hope you didn't miss me while I was on my first summer vacation in three years. So let's continue this series. If you remember, part 3 consisted mainly of the observation that FS-sets have a partial semigroup structure induced by \(\mathbb{F}\) as well as me telling you that there's a immediate correspondence between FS-sets and idempotent ultrafilters.

I'm slowly getting where I wanted to head all along with this series. When I write "my most lacking intuitions" in the title, I have my intuitions about central sets in mind. They are most lacking, I assure you. But with this series I wanted to clear my head a little. So let's head down the rabbit hole, no questions asked.

Towards the Central Sets Theorem

The Central Sets Theorem was conceived by Hillel Fürstenberg. I know relatively little about its history so I am still amazed by the fact that anyone could come up with it -- it's such a strange creature. Fürstenberg proved it for \(\mathbb{Z}\) (but I'll keep considering \(\mathbb{N}\), the two situations are equivalent anyway). The notion of centrality has its origins in ergodic theory -- unsurprisingly for Furstenberg. As fascinating and fruitful as the connection between ergodic Ramsey theory and Algebra in the Stone–Čech compactification is, I don't plan to introduce the technology in this series, because it will take us too far from the path I have in mind (mind you, I haven't even introduced the semigroup structure on \(\beta \mathbb{N}\), so really, I shouldn't introduce the ergodic point of view of which I know far less).

Fürstenberg devised the notion of central set (for subsets of \(\mathbb{N}\)) which was determined by recurrence phenomena in dynamical systems. Again, I don't want to discuss the dynamical point of view but I'll give the ultrafilter characterization later. The key for a connection to Ramsey theory was simple.

Theorem (Fürstenberg) Central sets (whatever they are) are partition regular.

Whatever central sets are (sorry for being temporarily mysterious), it is shocking how much algebraic structure central sets have to offer -- which is what the Central Sets Theorem is all about. It took a while and until Neil Hindman and Vitaly Bergelson made the connection between the algebraic/ultrafilter side and the ergodic side apparent. Nevertheless, the precise level of the algebraic closure of central sets is still a mystery. And this mystery is the reason for this series.

A detour: FP-sets and their condensations

One thing we should do before formulating the theorem is the following. What does an FS-set mean in the context of \(\mathbb{F}\)? Generally speaking, in a (partial) semigroup we have FP-sets ("finite products" instead of "finite sums"): given a sequence \((x_n: n\in \omega)\) we write

\[ FP(x_n) = { \prod_{i \in s} x_i: s \in \mathbb{F} }. \]

Of course, in the partial semigroup scenario, we should also think of this as a statement restricted to defined products. However, usually (e.g., in Hindman's Theorem) all products will be defined. Also, in the non-commutative case (which I wholeheartedly ignore in this series), this notation is supposed to be read "in order", i.e., the products are in the natural order of \(s \subseteq \omega\).

For a crucial example, consider \(\mathbb{F}\) itself.

If we have a sequence \((s_n : n\in \omega)\) in \(\mathbb{F}\) such that the \(s_n\) are pairwise disjoint, then the FP-set will be just fine -- all products are defined in our partial semigroup.

This is a critical example also because we can transfer such partial subsemigroups easily: If \(FP(s_n) \subseteq \mathbb{F}\) and we have some \(FS(x_n) \subseteq \mathbb{N}\), then we can consider \(y_n := \sum_{i\in s_n} x_i\). As long as the \(s_n\) are pairwise disjoint, we get

\[ FS(y_n) \subseteq FS(x_n).\]

So we have a partial subsemigroup of \(FS(x_n)\) induced by a partial subsemigroup of \(\mathbb{F}\)!

And this is actually a typical phenomenon thanks to Hindman's Theorem. Remember,

Hindman's Theorem If we partition an FS-set into finitely many pieces, one piece will contain an FS-set.

Condensations and Intuitions

There's one important question when it comes to developing an intuition: what should we expect when we partition again and again? One typical phenomenon is the following: if we take some \(FS(x_n)\) and partition into two pieces where one piece contains exactly the elements of the generating sequence \({ x_n : n \in \omega }\). By Hindman's Theorem, one part of the partition will contain an FS-set -- but that's not going to be \({ x_n : n \in \omega }\)! Consider the case \(x_n = 2^n\), then \({ 2^n : n \in \omega }\) certainly does not contain an FS-set, it does not even satisfy Schur's Theorem!

So what happens in this case? Well, we can easily describe many FS-sets in the other part of the partition; e.g., take every other generator and add: \(y_n= x_{2n} +x_{2n+1}\). Then \(FS(y_n)\) is good for the second part of the partition.

More generally, we could take any pairwise disjoint \((s_n : n\in \omega)\) in \(\mathbb{F}\), just make sure that no \(s_n\) is a singleton. Then as above, \(FP(s_n)\) induces an FS-subset of \(FS(x_n)\) -- which will lie completely in the "large" part of the partition.

A word of caution: in a certain sense, partitions as the one above are unusually simple because one part does not contain an FS-set. In general, you should expect all parts to contain FS-sets (for example, when separating different idempotent ultrafilters). Nevertheless, I would say that a huge chunk of arguments regarding strongly summable ultrafilters relies on such "simple" partitions -- so they are extremely useful.

The point I'm trying to make is that whenever we repeatedly partition an FS-set, Hindman's Theorem will give us homogeneous FS-sets -- but you should expect the elements in the generating sequence to be sums of many elements of the original generators!

This is why a sequence \((y_n: n \in \omega)\) in some \(FS(x_n)\) with \(FS(y_n)\subseteq FS(x_n)\) is called a condensation of \((x_n :n \in \omega)\) -- because generally speaking many elements from \(x_n\) are condensed into one of the \(y_n\) (of course, some \(x_n\) might just be dropped completely). The term "condensation" is also used in arbitrary (partial) semigroups.

Oh and to be absolutely clear:

Sequences in \(\mathbb{F}\) are always assumed to be pairwise disjoint (so that they are pairwise compatible in our partial operation)

Alright, that's enough for this part, I think. Next time, I'll finally talk about the Central Sets Theorem.