Erdős–Rado for arbitrary infinite cardinals and finite arity

For $n=0$, let $c:[{\kappa }^{+}{]}^1\to \kappa$. If every color fiber had size less than ${\kappa }^{+}$, each would have size at most $\kappa$ by the successor-cardinal definition. AC chooses an injection of each fiber into $\kappa$, giving an injection of their union into $\kappa \times \kappa$ by recording the color and its fiber index. F3 bounds the union by $\kappa$, contrary to its size ${\kappa }^{+}$. Therefore some fiber has size ${\kappa }^{+}$ and is homogeneous. By F1 this is exactly the required zero case.

Assume the assertion at $m\ge 0$, and let $F:[{\mathrm{\beth }}_{m+1}(\kappa {)}^{+}{]}^{m+2}\to \kappa$. Put $\mu ={\mathrm{\beth }}_m(\kappa )$. Repeatedly applying F6 to the recurrence F1 gives $\mu \ge \kappa$ and $\mu$ infinite. Thus F2 applies with $r=\kappa$, $d=m+1\ge 1$ and $\lambda =(2^{\mu }{)}^{+}={\mathrm{\beth }}_{m+1}(\kappa {)}^{+}$. Obtain distinct nodes $x_{\alpha }$ for $\alpha <{\mu }^{+}$ and $a$ outside their range.

Define $G:[{\mu }^{+}{]}^{m+1}\to \kappa$ by $G(v)=F(\{x_{\beta }:\beta \in v\}\cup \{a\})$. The argument has size $m+2$ because the $x_{\beta }$ are distinct and $a$ is outside their range, so $G$ is well-defined. The induction assertion at $m$, with $\mu ={\mathrm{\beth }}_m(\kappa )$, gives $H\subseteq {\mu }^{+}$ of size ${\kappa }^{+}$ homogeneous for $G$, of some color $i<\kappa$.

Put $Y=\{x_{\alpha }:\alpha \in H\}$; injectivity gives $\mathrm{\mid }Y\mathrm{\mid }={\kappa }^{+}$. For any $m+2$ nodes of $Y$, order their indices as ${\alpha }_0<\cdots <{\alpha }_{m+1}$ and put $u=\{x_{{\alpha }_0},\dots ,x_{{\alpha }_m}\}$. These are $m+1$ previous nodes at stage ${\alpha }_{m+1}$, so F2 gives $F(u\cup \{x_{{\alpha }_{m+1}}\})=F(u\cup \{a\})=G(\{{\alpha }_0,\dots ,{\alpha }_m\})=i$. Hence $Y$ is homogeneous. Only the largest index was used; no ambient increase of the $x_{\alpha }$ was assumed. This proves the successor step; F4 and step 1.1 prove all finite $n$.