 Solution

A theory is $\kappa$-satisfiable when every subtheory of size below $\kappa$ has a model. An uncountable cardinal $\kappa$ is weakly compact when every $\kappa$-satisfiable theory in an infinitary language $L_{\kappa ,\kappa }$ with at most $\kappa$ nonlogical symbols is satisfiable.

Equivalently, $\kappa \to (\kappa {)}_2^2$: every coloring $c:[\kappa {]}^2\to 2$ has a homogeneous subset of cardinality $\kappa$.