ASL North American Meeting, March 2012

I gave a special session talk at the ASL 2012 North American Annual Meeting (Madison, March 31–April 3, 2012).

Talk Title: The extent of the failure of Ramsey’s theorem at successor cardinals.

Extended abstract: Ramsey’s theorem asserts that for every coloring $c:[\omega {]}^2\to 2$, there exists an infinite subset $H\subseteq \omega$ such that $c\upharpoonright [H{]}^2$ is constant. At the early 1930’s, Sierpinski showed that a generalization of Ramsey’s theorem must fail for successor cardinals, and ever since the extent of this failure was studied extensively by many set-theorists, including, Erdos, Eisworth, Hajnal, Moore, Shelah, and Todorcevic.

In this talk, we shall show that Shelah’s notion of strong coloring ${\text{Pr}}_0({\lambda }^{+},{\lambda }^{+},{\lambda }^{+},\omega )$ coincides with the most basic concept considered already by Erdos and his collaborators: ${\lambda }^{+}\nrightarrow [{\lambda }^{+}{]}_{{\lambda }^{+}}^2$. More specifically, we shall discuss the following ZFC result.
Theorem. The following are equivalent for every uncountable cardinal $\lambda$:
(1) There exists a coloring $c:[{\lambda }^{+}{]}^2\to {\lambda }^{+}$ such that for every
(-) color $\gamma <{\lambda }^{+}$, and every
(-) subset $A\subseteq {\lambda }^{+}$ of size ${\lambda }^{+}$,
there exist $\alpha ,\beta \in A$  with $\alpha <\beta$ such that $$c(\alpha ,\beta )=\gamma .$$
(2) There exists a coloring $c:[{\lambda }^{+}{]}^2\to {\lambda }^{+}$ such that for every
(-) coloring $g:n\times n\to {\lambda }^{+}$ (here $n$ is a positive integer), and every
(-) family $\mathcal{A}\subseteq [{\lambda }^{+}{]}^n$ of size of ${\lambda }^{+}$ of mutually disjoint sets,
there exist $a,b\in \mathcal{A}$ with $max(a)<min(b)$ such that $$c(a_i,b_j)=g(i,j)\text{ for all }i,j<n.$$

(here, $a_i$ denotes the $i_{th}$-element of $a$, and $b_j$ denotes the $j_{th}$-element of $b$.)

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