 Erdős-Rado theorem for pairs

For every infinite cardinal $\lambda$, every coloring of pairs from $(2^{\lambda }{)}^{+}$ by $\lambda$ colors has a monochromatic subset of size ${\lambda }^{+}$. Construct an end-homogeneous tree by routing a new vertex through the least preceding vertices that match its colors to all previously chosen ancestors. An ancestor's color to every later vertex on a branch is constant. A node at height $\alpha$ is determined by its color sequence of length $\alpha$, so levels below ${\lambda }^{+}$ have size at most ${\lambda }^{|\alpha |}\le 2^{\lambda }$. The whole tree has more than $2^{\lambda }$ vertices, forcing a branch of length ${\lambda }^{+}$. One of the $\lambda$ ancestor-colors occurs ${\lambda }^{+}$ times, giving the required set.