Weakly compact cardinal

Summary

A weakly compact cardinal is an uncountable cardinal $\kappa$ for which the infinite Ramsey theorem holds at $\kappa$. This means that every colouring of the pairs of elements of $\kappa$ with two colours has a homogeneous set of size $\kappa$. Weakly compact cardinals are inaccessible and even Mahlo. The name comes from an equivalent property: a compactness theorem for the infinitary logic $\mathcal{L}_{\kappa, \kappa}$ in the case of theories with at most $\kappa$ sentences. Every measurable cardinal is weakly compact.