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Weakly compact cardinals lie at the focal point of a number of diverse concepts in infinite combinatorics, admitting various characterizations in terms of these concepts. If $\kappa^{ {<}\kappa} = \kappa$, then the following are equivalent:
Weak compactness
A cardinal $\kappa$ is weakly compact if and only if it is
uncountable
and every $\kappa$-satisfiable theory in an
$\mathcal{L}_{\kappa,\kappa}$
language of size at most $\kappa$ is satisfiable.
Extension property
A cardinal $\kappa$ is weakly compact if and only if for every
$A\subset V_\kappa$, there is a transitive structure $W$ properly
extending $V_\kappa$ and $A^*\subset W$ such that $\langle
V_\kappa,{\in},A\rangle\prec\langle W,{\in},A^*\rangle$.
Tree property
A cardinal $\kappa$ is weakly compact if and only if it is
inaccessible
and has the tree
property.
Filter property
A cardinal $\kappa$ is weakly compact if and only if whenever $M$ is a
set containing at most $\kappa$-many subsets of $\kappa$, then there
is a $\kappa$-complete nonprincipal
filter
$F$ measuring every set in $M$.
Weak embedding property
A cardinal $\kappa$ is weakly compact if and only if for every
$A\subset\kappa$ there is a transitive set $M$ of size $\kappa$ with
$\kappa\in M$ and a transitive set $N$ with an
embedding
$j:M\to N$ with
critical point
$\kappa$.
Embedding characterization
A cardinal $\kappa$ is weakly compact if and only if for every
transitive set $M$ of size $\kappa$ with $\kappa\in M$ there is a
transitive set $N$ and an embedding $j:M\to N$ with critical point
$\kappa$.
Normal embedding characterization
A cardinal $\kappa$ is weakly compact if and only if for every
$\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding
$j:M\to N$ with critical point $\kappa$, such that $N=\{
j(f)(\kappa)\mid f\in M\ \}$.
Hauser embedding characterization
A cardinal $\kappa$ is weakly compact if and only if for every
$\kappa$-model $M$ there is a $\kappa$-model $N$ and an embedding
$j:M\to N$ with critical point $\kappa$ such that $j,M\in N$.
Partition property
A cardinal $\kappa$ is weakly compact if and only if the partition
property
$\kappa\to(\kappa)^2_2$ holds.
Indescribability property
A cardinal $\kappa$ is weakly compact if and only if it is
$\Pi_1^1$-indescribable.
Skolem Property
A cardinal $\kappa$ is weakly compact if and only if $\kappa$ is
inaccessible and every $\kappa$-unboundedly satisfiable
$\mathcal{L}_{\kappa,\kappa}$-theory $T$ of size at most $\kappa$
has a model of size at least $\kappa$. A theory $T$ is
$\kappa$-unboundedly satisfiable if and only if for any
$\lambda<\kappa$, there exists a model $\mathcal{M}\models T$
with $\lambda\leq|M|<\kappa$. For more info see
here.
Weakly compact cardinals first arose in connection with (and were named for) the question of whether certain infinitary logics satisfy the compactness theorem of first order logic. Specifically, in a language with a signature consisting, as in the first order context, of a set of constant, finitary function and relation symbols, we build up the language of $\mathcal{L}_{\kappa,\lambda}$ formulas by closing the collection of formulas under infinitary conjunctions $\wedge_{\alpha<\delta}\varphi_\alpha$ and disjunctions $\vee_{\alpha<\delta}\varphi_\alpha$ of any size $\delta<\kappa$, as well as infinitary quantification $\exists\vec x$ and $\forall\vec x$ over blocks of variables $\vec x=\langle x_\alpha\mid\alpha<\delta\rangle$ of size less than $\kappa$. A theory in such a language is satisfiable if it has a model under the natural semantics. A theory is $\theta$-satisfiable if every subtheory consisting of fewer than $\theta$ many sentences of it is satisfiable. First order logic is precisely $L_{\omega,\omega}$, and the classical Compactness theorem asserts that every $\omega$-satisfiable $\mathcal{L}_{\omega,\omega}$ theory is satisfiable. A uncountable cardinal $\kappa$ is strongly compact if every $\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory is satisfiable. The cardinal $\kappa$ is weakly compact if every $\kappa$-satisfiable $\mathcal{L}_{\kappa,\kappa}$ theory, in a language having at most $\kappa$ many constant, function and relation symbols, is satisfiable.
Next, for any cardinal $\kappa$, a $\kappa$-tree is a tree of height $\kappa$, all of whose levels have size less than $\kappa$. More specifically, $T$ is a tree if $T$ is a partial order such that the predecessors of any node in $T$ are well ordered. The $\alpha^{\rm th}$ level of a tree $T$, denoted $T_\alpha$, consists of the nodes whose predecessors have order type exactly $\alpha$, and these nodes are also said to have height $\alpha$. The height of the tree $T$ is the first $\alpha$ for which $T$ has no nodes of height $\alpha$. A “”$\kappa$-branch”” through a tree $T$ is a maximal linearly ordered subset of $T$ of order type $\kappa$. Such a branch selects exactly one node from each level, in a linearly ordered manner. The set of $\kappa$-branches is denoted $[T]$. A $\kappa$-tree is an Aronszajn tree if it has no $\kappa$-branches. A cardinal $\kappa$ has the tree property if every $\kappa$-tree has a $\kappa$-branch.
A transitive set $M$ is a $\kappa$-model of set theory if $|M|=\kappa$, $M^{\lt\kappa}\subset M$ and $M$ satisfies ZFC$^-$, the theory ZFC without the power set axiom (and using collection and separation rather than merely replacement). For any infinite cardinal $\kappa$ we have that $H_{\kappa^+}$ models ZFC$^-$, and further, if $M\prec H_{\kappa^+}$ and $\kappa\subset M$, then $M$ is transitive. Thus, any $A\in H_{\kappa^+}$ can be placed into such an $M$. If $\kappa^{\lt\kappa}=\kappa$, one can use the downward Löwenheim-Skolem theorem to find such $M$ with $M^{\lt\kappa}\subset M$. So in this case there are abundant $\kappa$-models of set theory (and conversely, if there is a $\kappa$-model of set theory, then $2^{\lt\kappa}=\kappa$).
The partition property $\kappa\to(\lambda)^n_\gamma$ asserts that for every function $F:[\kappa]^n\to\gamma$ there is $H\subset\kappa$ with $|H|=\lambda$ such that $F\upharpoonright[H]^n$ is constant. If one thinks of $F$ as coloring the $n$-tuples, the partition property asserts the existence of a monochromatic set $H$, since all tuples from $H$ get the same color. The partition property $\kappa\to(\kappa)^2_2$ asserts that every partition of $[\kappa]^2$ into two sets admits a set $H\subset\kappa$ of size $\kappa$ such that $[H]^2$ lies on one side of the partition. When defining $F:[\kappa]^n\to\gamma$, we define $F(\alpha_1,\ldots,\alpha_n)$ only when $\alpha_1<\cdots<\alpha_n$.
Every weakly compact cardinal is weakly compact in $L$. (Jech, 2003)
Nevertheless, the weak compactness property is not generally downward absolute between transitive models of set theory.
To expand using [2]
An inaccessible cardinal $κ$ is $Σ_n$-weakly compact iff it reflects $Π_1^1$ sentences with $Σ_n$-predicates, i.e. for every $R ⊆ V_κ$ which is definable by a $Σ_n$ formula (with parameters) over $V_κ$ and every $Π_1^1$ sentence $Φ$, if $\langle V_κ , ∈, R \rangle \models Φ$ then there is $α < κ$ (equivalently, unboundedly-many $α < κ$) such that $\langle V_α , ∈, R ∩ V_α \rangle \models Φ$. Analogously for $Π_n$ and $∆_n$. $κ$ is $Σ_ω$-weakly compact iff it is $Σ_n$-weakly compact for all $n < ω$.
$κ$ is $Σ_n$-weakly compact $\iff$ $κ$ is $Π_n$-weakly compact $\iff$ $κ$ is $∆_{n+1}$-weakly compact $\iff$ For every $Π_1^1$ formula $Φ(x_0 , …, x_k)$ in the language of set theory and every $a_0 , …, a_k ∈ V_κ$, if $V κ \models Φ(a_0 , …, a_k )$, then there is $λ ∈ I_n := \{λ < κ : λ$ is inaccessible and $V_λ \preccurlyeq_n V_κ\}$ such that $V_λ \models Φ(a_0 , …, a_k)$.
In (Bosch, 2006) it is shown that every $Σ_ω$-w.c. cardinal is $Σ_ω$-Mahlo and the set of $Σ_ω$-Mahlo cardinals below a $Σ_ω$-w.c. cardinal is $Σ_ω$-stationary, but if κ is $Π_{n+1}$-Mahlo, then the set of $Σ_n$-w.c. cardinals below $κ$ is $Π_{n+1}$-stationary.
These properties are connected with some forms of absoluteness. For example, the existence of a $Σ_ω$-w.c. cardinal is equiconsistent with the generic absoluteness axiom $\mathcal{A}(L(\mathbb{R}), \underset{\sim}{Σ}_ω , Γ)$ where $Γ$ is the class of projective ccc forcing notions.
This section from (Leshem, 2000; Bagaria, 2006)