Weakly compact cardinal

Remarks

The cardinal $\aleph_0$ satisfies clause 2 by Ramsey's theorem, and it fails only clause 1. The cardinal $\aleph_1$ fails clause 2: Sierpiński's colouring compares the order of $\omega_1$ with the order of an injective family of real numbers, and it has no uncountable homogeneous set. Equivalent form: $\kappa$ is weakly compact if and only if $\kappa$ is inaccessible and has the tree property. The tree property says that every tree of height $\kappa$ whose levels have size less than $\kappa$ has a branch of length $\kappa$. Another equivalent form: $\kappa$ is inaccessible, and a compactness theorem holds for the infinitary logic $\mathcal{L}_{\kappa, \kappa}$. It says that a set $\Sigma$ of at most $\kappa$ sentences has a model whenever every subset of $\Sigma$ of size less than $\kappa$ has one.