November 22
Brent Cody, Virginia Commonwealth University
A refinement of the Ramsey hierarchy via indescribability
A subset $X$ of a cardinal $\kappa$ is Ramsey if for every function $f:[X]^{<\omega}\rightarrow 2$ there is a set $H\subseteq X$ of cardinality $\kappa$ which is homogeneous for $f$, meaning that $f\upharpoonright[H]^n$ is constant for each $n<\omega$. Baumgartner proved that if $\kappa$ is Ramsey, then the collection of non-Ramsey subsets of $\kappa$ is a normal ideal on $\kappa$. We will discuss some recent results concerning Ramsey properties in which homogeneous sets are demanded to be indescribable of a particular degree. Moreover, by iterating Feng's Ramsey operator, which he used to define a notion of $\alpha$-Ramseyness of a cardinal $\kappa$, we will consider hypotheses in which homogeneous sets themselves satisfy various Ramsey properties. For ordinals $\alpha,\beta<\kappa$ we will define a notion of $\alpha$-$\Pi^1_\beta$-Ramseyness of a cardinal $\kappa$ where $\alpha$ indicates how many times the Ramsey operator has been iterated and $\beta$ indicates the degree of transfinite indescribability (due to Sharpe-Welch and independently Bagaria) one initially demands homogeneous to satisfy. We will prove that for $\alpha,\beta<\kappa$ an $\alpha$-$\Pi^1_\beta$-Ramsey cardinal is strictly between Feng's $\alpha$-Ramsey and an ($\alpha+1$)-Ramsey cardinal in consistency strength. Moreover, for fixed $\alpha<\kappa$, as $\beta$ increases the $\alpha$-$\Pi^1_\beta$-Ramsey cardinals yield a strictly increasing hierarchy, in a somewhat subtle sense. For $\beta_0<\beta_1<\kappa$ and for large enough $\alpha<\kappa$, $\kappa$ being $\alpha$-$\Pi^1_{\beta_0}$-Ramsey is equivalent to $\kappa$ being $\alpha$-$\Pi^1_{\beta_1}$-Ramsey (we will identify the least $\alpha$ at which this equivalence occurs). But if $\alpha,\beta_0<\kappa$ there is a large enough $\beta_1<\kappa$ such that $\kappa$ being $\alpha$-$\Pi^1_{\beta_0}$-Ramsey is strictly weaker than $\kappa$ being $\alpha$-$\Pi^1_{\beta_1}$-Ramsey. All of these results seem to require a careful analysis of the ideals associated to the various large cardinal notions.