---
tag: 4F1ABY39W1
kind: definition
name: Weakly compact cardinal
msc: ["03E55", "03E02"]
defines: 1RHTVMH34G
uses: [0EBACQFAZW, 0W7FBYS1F1, NW4DREQ5DQ]
sources:
  - url: https://en.wikipedia.org/wiki/Weakly_compact_cardinal
---

## Definition

Let $\kappa$ be a [[cardinal|0EBACQFAZW]], and write $[\kappa]^2$ for the set of two-element subsets of $\kappa$.
The cardinal $\kappa$ is *weakly compact* if:

1. $\kappa$ is [[uncountable|0W7FBYS1F1]].
2. For every function $c \colon [\kappa]^2 \to \set{0, 1}$ there is a set $H \subseteq \kappa$ with $\abs{H} = \kappa$ such that $c$ is constant on $[H]^2$.

Clause 2 is written $\kappa \to (\kappa)^2_2$.

## 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|NW4DREQ5DQ]] 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.