What Is the Definition of a Weakly Compact Cardinal?

A weakly compact cardinal is a large cardinal defined by a combination of strong size conditions and a reflection property. The most common formulations use a partition relation or the tree property. These formulations are equivalent, but the equivalence is a theorem; it is useful to know which definition a text is using.

The standard definition

A cardinal $\kappa$ is weakly compact if it is strongly inaccessible and satisfies the partition relation

$$\kappa \to (\kappa)^2_2.$$

Strong inaccessibility means that $\kappa$ is uncountable, regular, and a strong limit cardinal. Regularity says that no set of fewer than $\kappa$ smaller ordinals is cofinal in $\kappa$. The strong limit condition says $2^\lambda<\kappa$ for every cardinal $\lambda<\kappa$.

The partition relation describes a coloring problem. The notation means that for every function

$$c:[\kappa]^2\to\{0,1\},$$

there is a set $H\subseteq\kappa$ of size $\kappa$ such that every pair from $H$ receives the same color. Here $[\kappa]^2$ is the collection of two-element subsets of $\kappa$. In other words, no matter how you color all pairs using two colors, there is a monochromatic subset as large as $\kappa$.

Some references present the partition property as the definition, with uncountability understood or stated separately. The strong inaccessibility condition is part of the standard definition and should not be silently omitted when comparing formulations.

The equivalent tree-property definition

A $\kappa$-tree is a partially ordered set arranged in levels indexed by ordinals below $\kappa$, with each level having size less than $\kappa$. A cofinal branch is a linearly ordered path meeting every level. The tree property at $\kappa$ says that every $\kappa$-tree has a cofinal branch.

For an inaccessible cardinal $\kappa$, the following are equivalent:

• $\kappa$ is weakly compact.
• Every coloring $[\kappa]^2\to\{0,1\}$ has a homogeneous subset of size $\kappa$.
• Every $\kappa$-tree has a cofinal branch.

Thus many authors define weak compactness as inaccessibility plus the tree property instead of using the partition relation. The tree formulation is often helpful when studying structures built level by level; the partition formulation is useful in Ramsey theory.

Another standard equivalent formulation is $\Pi^1_1$ indescribability. Roughly, it says that certain second-order properties true in the universe of sets below $\kappa$ must already hold in a smaller such universe. This expresses a strong form of reflection. The precise statement requires a formal definition of the relevant logical language, so it is usually introduced separately.

A small example that clarifies the notation

The partition relation by itself does not identify weak compactness unless the size and inaccessibility conditions are included. For example, the infinite Ramsey theorem gives

$$\omega\to(\omega)^2_2.$$

That says every two-coloring of pairs of natural numbers has an infinite monochromatic subset. But $\omega$ is not uncountable, so it is not strongly inaccessible and is not weakly compact.

For a hypothetical uncountable weakly compact $\kappa$, the statement is much stronger: every two-coloring of pairs from a set of size $\kappa$ must contain a monochromatic set of size $\kappa$. It is not enough to find a monochromatic set that is merely infinite or unbounded.

What the definition does not mean

“Weakly compact” is not a claim about compactness in the usual topological sense. The terminology is connected with compactness properties in infinitary logic, but weak compactness has several precise set-theoretic characterizations, including the ones above.

Also, being inaccessible alone does not make a cardinal weakly compact. Inaccessibility supplies the size conditions; the partition, tree, or reflection property is the additional large-cardinal strength.

Common mistakes

• Leaving out inaccessibility: A partition statement at a small cardinal, such as $\omega$, does not make that cardinal weakly compact.
• Reading the arrow too loosely: $\kappa\to(\kappa)^2_2$ concerns every coloring of pairs and requires a monochromatic subset of size exactly $\kappa$.
• Treating the tree property as a definition for every cardinal: The standard equivalence is stated for inaccessible $\kappa$. Keep that hypothesis when using the tree formulation.
• Confusing the term with topological compactness: The relevant compactness ideas are logical and set-theoretic, not simply compactness of a topological space.