# 6th European Set Theory Conference, July 2017

I gave a 3-lecture tutorial at the 6th European Set Theory Conference in Budapest, July 2017.

Title: Strong colorings and their applications.

Abstract. Consider the following questions.

• Is the product of two $\kappa$-cc partial orders again $\kappa$-cc?
• Does there exist a regular hereditary separable topological space which is non-Lindelof?
• Given an $\aleph_1$-sized Abelian group $(G,+)$, must there exist a unary function $f:G\rightarrow G$ such that any proper substructure of $(G,+,f)$ be countable?

It turns out that all of the above questions can be decided (in one way), provided that there exists a certain “strong coloring” (or “wild partition”) of a corresponding uncountable graph.
In this tutorial, we shall present some of the techniques involved in constructing such strong colorings, and demonstrate how partial orders/topological spaces/algebraic structures may be derived from these colorings.

Lecture 1 ** Lecture 2 ** Lecture 3

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### 4 Responses to 6th European Set Theory Conference, July 2017

1. Literature says:

I was surprised to see on page 19 of Lecture 2 a result showing $\aleph_2\nrightarrow[\aleph_1]^3_{\aleph_1}$ from some hypothesis when in the literature one can find a result showing that this negative square-bracket partition relation is in fact equivalent to the negation of Chang’s Conjecture.

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• saf says:

Indeed, the equivalency between this and the failure of Chang’s conjecture is proved in (Todorcevic, 1994) which is cited on the very same page 19.
I presented Shore’s argument (from 1974) just to demonstrate how Kurepa trees may be used to lift strong colorings of dimension $n$ to strong colorings of dimension $n+1$.

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• Literature says:

I see no such citation on page 19. Maybe it is somewhere else?
Regarding the theorem on top of page 19 that is credited to Shore 1974 after consulting the paper of Shore I note that there is no proof of this result there but rather the result where the assumption of the existence of a Luzin set is replaced by the stronger assumption of CH. What Shore shows that Hajnal’s stepping-up method via the existence of a Kurepa tree allows one to step-up the negative square-bracket relation $\aleph_1\nrightarrow[\aleph_0 : \aleph_1]^2_{\aleph_1}$ to $\aleph_2\nrightarrow[\aleph_1]^3_{\aleph_1}$. Shore does not show that the existence of a Luzin set implies the partition relation $\aleph_1\nrightarrow[\aleph_0 : \aleph_1]^2_{\aleph_1}$. This result appears in the literature only in 1987. Today we know a general stepping up method that applies to arbitrary negative square bracket partition relations $\kappa\nrightarrow[\lambda]^n_\theta$ with $\lambda$ regular uncountable. It is pity that the Lecture 2 does not mention this.

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• saf says:

The citation is there – just skip to the very last page of the PDF.

Yes, Shore’s argument is the stepping-up $\aleph_1\nrightarrow[\aleph_0 : \aleph_1]^2_{\aleph_1}$ to $\aleph_2\nrightarrow[\aleph_1]^3_{\aleph_1}$ using a Kurepa tree. I bet Hajnal knew this argument as well, but in the Erdos-Hajnal-Mate-Rado book (Theorem 51.2), this is attributed to (Shore, 1974).

Generally speaking, the challenge of this tutorial is to deliver as many ideas as possible while keeping a reasonable flow. To obtain such a flow, one sometimes needs to connect disjoint components. In any case, I have updated the slides to include a citation on page 14, and changed the hypothesis of Shore’s theorem to better reflect what was known back in 1974.

There are of course many additional ideas that did not make it into this 3-lectures tutorial. I may expend these slides in the future, but not before I am done with a couple of previous commitments.

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