Erdős cardinal

In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958).

A cardinal ${\displaystyle \kappa }$ is called ${\displaystyle \alpha }$-Erdős if for every function ${\displaystyle f:[\kappa ]^{<\omega }\to \{0,1\}}$, there is a set of order type ${\displaystyle \alpha }$ that is homogeneous for ${\displaystyle f}$. In the notation of the partition calculus, ${\displaystyle \kappa }$ is ${\displaystyle \alpha }$-Erdős if

${\displaystyle \kappa \rightarrow (\alpha )^{<\omega }}$.

Under this definition, any cardinal larger than the least ${\displaystyle \alpha }$-Erdős cardinal is ${\displaystyle \alpha }$-Erdős.

The existence of zero sharp implies that the constructible universe ${\displaystyle L}$ satisfies "for every countable ordinal ${\displaystyle \alpha }$, there is an ${\displaystyle \alpha }$-Erdős cardinal". In fact, for every indiscernible ${\displaystyle \kappa }$, ${\displaystyle L_{\kappa }}$ satisfies "for every ordinal ${\displaystyle \alpha }$, there is an ${\displaystyle \alpha }$-Erdős cardinal in ${\displaystyle \mathrm {Coll} (\omega ,\alpha )}$" (the Lévy collapse to make ${\displaystyle \alpha }$ countable).

However, the existence of an ${\displaystyle \omega _{1}}$-Erdős cardinal implies existence of zero sharp. If ${\displaystyle f}$ is the satisfaction relation for ${\displaystyle L}$ (using ordinal parameters), then the existence of zero sharp is equivalent to there being an ${\displaystyle \omega _{1}}$-Erdős ordinal with respect to ${\displaystyle f}$. Thus, the existence of an ${\displaystyle \omega _{1}}$-Erdős cardinal implies that the axiom of constructibility is false.

The least ${\displaystyle \omega }$-Erdős cardinal is not weakly compact,^{[1]p. 39.} nor is the least ${\displaystyle \omega _{1}}$-Erdős cardinal.^{[1]p. 39}

If ${\displaystyle \kappa }$ is ${\displaystyle \alpha }$-Erdős, then it is ${\displaystyle \alpha }$-Erdős in every transitive model satisfying "${\displaystyle \alpha }$ is countable."

Dodd's Notion of Erdős Cardinals

For a limit ordinal ${\displaystyle \alpha }$, a cardinal ${\displaystyle \kappa }$ is less often called ${\displaystyle \alpha }$-Erdős if for every closed unbounded ${\displaystyle C\subseteq \kappa }$ and every function ${\displaystyle f:[C]^{<\omega }\rightarrow \kappa }$ such that ${\displaystyle f(x)<min(x)}$ for all ${\displaystyle x\in [C]^{<\omega }}$, there is a set ${\displaystyle H\subseteq C}$ of order-type ${\displaystyle \alpha }$ that is homogeneous for ${\displaystyle f}$.^{[2]p. 138.}

An equivalent definition is that ${\displaystyle \kappa }$ is ${\displaystyle \alpha }$-Erdős if for any ${\displaystyle A\subseteq \kappa }$, there is a set ${\displaystyle I}$ of order-type ${\displaystyle \alpha }$ of order-indiscernibles for the structure ${\displaystyle (L_{\kappa }[A];\in ,A)}$ such that:

• for every ${\displaystyle \beta \in I}$, ${\displaystyle (L_{\beta }[A];\in ,A)\prec (L_{\kappa }[A];\in ,A)}$, and
• for every ${\displaystyle \gamma <\kappa }$, the set ${\displaystyle I\setminus \gamma }$ forms a set of order-indiscernibles for the structure ${\displaystyle (L_{\kappa }[A];\in ,A,\xi )_{\xi <\gamma }}$.

The least cardinal ${\displaystyle \kappa }$ to satisfy the partition relation ${\displaystyle \kappa \rightarrow (\alpha )^{<\omega }}$ is still ${\displaystyle \alpha }$-Erdős under this definition. Every ${\displaystyle \omega }$-Erdős cardinal is an inaccessible limit of ineffable cardinals.^[3]

See also

• List of large cardinal properties