Erdős-Rado theorem for pairs ($(2^{\lambda }{)}^{+}⟶({\lambda }^{+}{)}_{\lambda }^2$)

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.

The colors $\nu$ here range over nonzero cardinals. The Erdős-Rado theorem for pairs makes this function total, and target and color monotonicity make it nondecreasing, with $f(\alpha )\ge \alpha$. If an uncountable cardinal $\kappa$ is closed below itself under $f$, then it is strong limit. Indeed, coloring distinct binary strings of length $\lambda <\kappa$ by their first differing coordinate uses $\lambda$ colors and has no monochromatic triangle, so $f(\lambda +1)>2^{\lambda }$. If $\kappa$ also has the tree property, the end-homogeneous tree for any $\nu <\kappa$ has levels of size below $\kappa$ and a cofinal branch; regularity makes one color occur $\kappa$ times. Thus $f(\kappa )=\kappa$. This distinguishes function closure plus the tree property from the tree property alone.