A question on a set theoretic theorem 
Erdős-Rado theorem: Let $\kappa$ be an infinite cardinal. Let $E$ be a set with $|E|>2^\kappa$, and suppose $[E]^2=\bigcup_{\alpha<\kappa}P_\alpha$. Then there exists $\alpha<\kappa$ and a subset $A$ of $E$ with $|A|>\kappa$ such that $[A]^2\subset P_\alpha$.

In this theorem, does there exist $\alpha < \kappa$, such that $|P_\alpha| > 2^\kappa$? If it is true, could $A$ (in theorem) be made more large, for example, $>2^\kappa$?
Thanks ahead.
 A: No such strengthening is possible in $\mathsf{ZFC}$. Suppose that $2^\kappa=\kappa^+$ and $2^{\kappa^+}=\kappa^{++}$. Let $E={}^{\kappa^+}2$, the set of functions from $\kappa^+$ to $2=\{0,1\}$. Let $\le$ be the lexicographic order on $E$, and let $\preceq$ be any well-ordering of $E$. Let $$P_0=\left\{\{f,g\}\in[E]^2:\,\le\text{ and }\preceq\text{ agree on }\{f,g\}\right\}$$ and $$P_1=\left\{\{f,g\}\in[E]^2:\,\le\text{ and }\preceq\text{ disagree on }\{f,g\}\right\}\;.$$ Clearly $|E|=2^{(\kappa^+)}=\kappa^{++}>\kappa^+=2^\kappa$, and $\{P_0,P_1\}$ is a partition of $[E]^2$.
Suppose that $A\subseteq E$ and $[A]^2\subseteq P_i$ for some $i\in 2$. There are an ordinal $\alpha$ and an enumeration $A=\{f_\xi:\xi<\alpha\}$ of $A$ such that $f_\xi\preceq f_\eta$ iff $\xi\le\eta$. Then $\langle f_\xi:\xi<\alpha\rangle$ is either a strictly increasing (if $i=0$) or a strictly decreasing (if $i=1$) $\alpha$-sequence in $\langle E,\le\rangle$. But any monotone $\kappa^{++}$-sequence in $\langle E,\le\rangle$ is eventually constant, so $\alpha<\kappa^{++}$, and $|A|\le\kappa^+=2^\kappa$. Thus, for this partition there is no homogeneous set of cardinality larger than $2^\kappa$.
