The fourteen Victoria Delfino problems and their status in the year 2019

February 1, 2019

The Cabal seminar in southern California was instrumental to the development of determinacy. The Delfino problems were suggested as a way to measure progress on this area. Fourteen problems were suggested in total through the years. Some were solved very quickly after their proposal, a few remain open.

Benedikt Löwe and I wrote a survey of their current status, cleverly titled The fourteen Victoria Delfino problems and their status in the year 2019. The cleverness has forced us to keep changing its title as its publication date kept being postponed. It is scheduled to appear in the fourth volume of the reissued Cabal volumes, which I am told is expected to finally be published this year. The volumes are being published by the Association for Symbolic Logic and Cambridge University Press as part of the Lecture Notes in Logic series.

The survey can be accessed through the Hamburger Beiträge zur Mathematik preprint server; it is paper 770 there. It can also be found through my papers page (currently under notes, and later on, once it appears, under papers).

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580 -Partition calculus (2)

April 1, 2009

1. Infinite exponents

Last lecture we showed that {\omega\rightarrow(\omega)^m_n} for any {m,n<\omega.} Later, we will see that for any {\lambda,\rho} and any {m<\omega} there is {\kappa} such that {\kappa\rightarrow(\lambda)^m_\rho.} On the other hand (recall we are assuming choice), there are no nontrivial positive partition relations {\kappa\rightarrow(\lambda_i)^\tau_{i<\rho}} with {\tau} infinite.

(To avoid vacuously true statements, we are always assuming implicitly that {\kappa\rightarrow(\lambda_i)^\tau_{i<\rho}} requires {\kappa\ge\lambda_i\ge\tau} for at least one {i<\rho.})

Theorem 1 ({\mbox{Erd\H os}}-Rado) For all {\kappa} and all infinite {\tau,} {\kappa\not\rightarrow(\tau)^\tau.}

 
Proof: It is enough to show that {\kappa\not\rightarrow(\aleph_0)^{\aleph_0}.} In effect, if {\kappa\rightarrow(\tau)^\tau} holds and {\tau>\aleph_0,} we may assume that {\kappa>\tau} is regular, so any countable subset {X} of {\kappa} is bounded, and the ordinal {\sup(X)+\tau} is still strictly below {\kappa.}

For {x\in[\kappa]^\tau,} let {{}_\omega x} denote the subset of {x} consisting of its first {\omega} many members.

Now, given {f:[\kappa]^{\aleph_0}\rightarrow2,} let {g:[\kappa]^\tau\rightarrow2} be the function {g(x)=f({}_\omega x).} If {H} is homogeneous for {g} and of size {\tau,} then {{}_\omega H} is homogeneous for {f.} In effect, let {H'=H\setminus{}_\omega H,} so {|H'|=\tau.} If {x\in[{}_\omega H]^{\aleph_0},} then {f(x)=g(x\cup H')=g(H),} by homogeneity of {H.}

This shows that {\kappa\rightarrow(\tau)^\tau} for {\tau} infinite implies that there is some {\kappa'} such that {\kappa'\rightarrow(\aleph_0)^{\aleph_0},} so it is enough to show that the latter never holds. In fact, we cannot even have {\kappa\rightarrow(\omega)^\omega.} (Recall our convention that {\omega} denotes the order type and {\aleph_0} the size.)

For this, let {\kappa} be an arbitrary infinite cardinal, and let {<} be a well-ordering of {[\kappa]^\omega.} Define {f:[\kappa]^\omega\rightarrow 2} by {f(s)=0} iff {s} is the {<}-least member of {[s]^\omega,} and {f(s)=1} otherwise.

If {x\in[\kappa]^\omega} is homogeneous for {f} and {y} is the {<}-least member of {[x]^\omega,} then {f(y)=0,} so {x} is {0}-homogeneous. Now consider any infinite sequence {x_0\subsetneq x_1 \subsetneq\dots\subseteq x} of infinite subsets of {x.} Since {f(x_n)=0} for all {n,} we must have that {\dots<x_1<x_0,} contradicting that {<} is a well-order. \Box

In the presence of the axiom of choice, Theorem 1 completely settles the question of what infinite exponent partition relations hold. Without choice, the situation is different.

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