Mini-conference in Set Theory

Fields Institute, room 230.
Schedule:
  1. 1:30-3:00 Jean A. Larson, University of Florida: Coloring Paths in the Random Graph
  2. 3:30-5:10 Justin Tatch Moore, Boise State University:
    1. 3:30-4:00 Shelah's conjecture.
    2. 4:05-4:35 The L space problem.
    3. 4:40-5:10 $\omega_1$ and $\omega_1^*$ may be the only minimal uncountable order types.
  3. 5:20-6:50 Uri Abraham, Carnegie Mellon University and Ben Gurion University: Polychromatic partition relations
Abstracts
Uri Abraham
Polychromatic partition relations
A function f from the unordered pairs of \lambda is k-bounded iff there are no k pairs with the same color. A subset X of \lambda is polychromatic iff all pairs from X have different colors. We study the existence of large polychromatic sets for arbitrary 2-bounded colorings. (This is joint work with James Cummings and C. Smyth.)
Justin Moore.
Shelah's Conjecture.
I will give an overview of the proof that Shelah's conjectured five element basis for the uncountable linear orders follows from PFA. The proof involves some new ingredients which are now known to be --- to a certain extent --- necessary. I will discuss why some of the more conventional approaches can't resolve Shelah's conjecture.
The L space problem.
I will present an overview of what is involved in constructing a ZFC L space. A "lite" version of this theorem is the negation of the rectangle hypothesis in ZFC. A proof of this weak version will be discussed in some more detail.
$\omega_1$ and $\omega_1^*$ may be the only minimal uncountable order types.
Both $\omega_1$ and its converse have the property that they are embeddable in all of their uncountable suborders. Consistently, there are other uncountable linear orders with this property --- Shelah's conjecture implies that there are five, for instance. Under CH, however, a classical result of Sierpinski forbids there from being any real types which are minimal. Baumgartner asked whether it was consistent that $\omega_1$ and $\omega_1^*$ were the only minimal uncountable types. His construction of a minimal Specker type assuming $\diamondsuit^+$ raised the possibility that there might be a ZFC construction of a new uncountable minimal type. I will show that this is not the case. This is done by demonstrating that a form of ladder system uniformization relativized to subtrees of an Aronszajn tree is consistent with CH but implies $2^{\aleph_0} = 2^{\aleph_1}$ in the presence of a minimal Specker type.

Toronto Set Theory Seminar
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