#definition #large_cardinals # Definitions $\kappa$ is weakly compact if one of the following equivalent definitions hold: 1. $\kappa \to (\kappa)^{2}_{\lambda}$ for every $\lambda<\kappa$. [[Infinite combinatorics#^909edc|I.e.]] For every coloring of $[\kappa]^{2}$ into lt;\kappa$ many colors, there is a homogeneous set of size $\kappa$. 2. $\kappa$ is inaccessible and has the tree property, i.e. there is no $\kappa$-[[Trees#^aron|Aronszajn tree]]. 3.