## Luminy – Coda

October 27, 2010

While at Luminy, David Asperó showed me a quick proof of a nice result on Reinhardt cardinals in ${\sf ZF}$. It complements Grigor Sargsyan’s result discussed here.

Theorem (Asperó). Work in ${\sf ZF}$. Suppose $j:V\to V$ is a nontrivial elementary embedding. Then there are a $\bar\kappa<{\rm cp}(j)$ and an ordinal $\alpha$ such that for all $\beta$ there is a $\mu$ and an elementary

$\pi:V_\mu\to V_\mu$

such that ${\rm cp}(\pi)=\bar\kappa$ and $\pi(\alpha)>\beta$.

Proof. For $\alpha$ an ordinal, set

$\kappa^\alpha=\min\{\kappa\mid\exists\mu\exists i:V_\mu\to V_\mu$ such that ${\rm cp}(i)=\kappa$ and ${\rm ot}(\{\beta<\mu\mid i(\beta)=\beta\})\ge\alpha\}$.

Note that suitable fragments of $j$ witness that $\kappa^\alpha$ is defined for all $\alpha$. Moreover, $\alpha<\beta$ implies that $\kappa^\alpha\le\kappa^\beta\le{\rm cp}(j)$, and therefore there is a $\bar\kappa\le{\rm cp}(j)$ such that $\kappa^\beta=\bar\kappa$ for all $\beta$ sufficiently large. Moreover, since it is definable, we actually have $\bar\kappa<{\rm cp}(j)$.

Let $\alpha$ be least with $\kappa^\beta=\bar\kappa$ for $\beta\ge\alpha$. We claim that $\bar\kappa$ and $\alpha$ are as wanted. For this, consider some $\beta>\alpha$, and pick $i:V_\mu\to V_\mu$ witnessing that $\bar\kappa=\kappa^\beta$. All we need to do is to check that $i(\alpha)\ge\beta$.

But note that if $\gamma\in[\alpha,\beta)$, then $V_\mu\models\kappa^\gamma=\bar\kappa$ Hence, if $i(\alpha)<\beta$, we have

$V_\mu\models \kappa^{i(\alpha)}=\bar\kappa$.

But $\kappa^{i(\alpha)}=i(\kappa^\alpha)=i(\bar\kappa)>\bar\kappa$. Contradiction. $\Box$

## 580 -Partition calculus (5)

April 21, 2009

1. Larger cardinalities

We have seen that ${\omega\rightarrow(\omega)^n_m}$ (Ramsey) and ${\omega\rightarrow[\omega]^n_\omega}$ (${\mbox{Erd\H os}}$-Rado) for any ${n,m<\omega.}$ On the other hand, we also have that ${2^\kappa\not\rightarrow(3)^2_\kappa}$ (${\mbox{Sierpi\'nski}}$) and ${2^\kappa\not\rightarrow(\kappa^+)^2}$ (${\mbox{Erd\H os}}$-Kakutani) for any infinite ${\kappa.}$

Positive results can be obtained for larger cardinals than ${\omega}$ if we relax the requirements in some of the colors. A different extension, the ${\mbox{Erd\H os}}$-Rado theorem, will be discussed later.

Theorem 1 (${\mbox{Erd\H os}}$-Dushnik-Miller) For all infinite cardinals ${\lambda,}$ ${\lambda\rightarrow(\lambda,\omega)^2.}$

This was originally shown by Dushnik and Miller in 1941 for ${\lambda}$ regular, with ${\mbox{Erd\H os}}$ providing the singular case. For ${\lambda}$ regular one can in fact show something stronger:

Theorem 2 (${\mbox{Erd\H os}}$-Rado) Suppose ${\kappa}$ is regular and uncountable. Then
$\displaystyle \kappa\rightarrow_{top}(\mbox{Stationary},\omega+1)^2,$ which means: If ${f:[\kappa]^2\rightarrow2}$ then either there is a stationary ${H\subseteq\kappa}$ that is ${0}$-homogeneous for ${f}$, or else there is a closed subset of ${\kappa}$ of order type ${\omega+1}$ that is ${1}$-homogeneous for ${f}$.

(Above, top stands for “topological.”)