[proofplan] The proof assembles two precise external characterization theorems for strongly inaccessible cardinals. The first is the Erdős-Tarski combinatorial characterization: for a strongly inaccessible cardinal $\kappa$, the tree property is equivalent to $\kappa \to (\kappa)^2_2$. The second is the Keisler-Makkai model-theoretic characterization: for a strongly inaccessible cardinal $\kappa$, the tree property is equivalent to $\kappa$-compactness of $L_{\kappa,\kappa}$ for languages and theories of size at most $\kappa$. After verifying that the hypotheses and size conventions of these two external results match the present statement, transitivity of equivalence gives the desired three-way equivalence. [/proofplan] [step:Apply the Erdős-Tarski characterization to compare the tree property and the partition relation] We invoke the Erdős-Tarski combinatorial characterization of weak compactness: if $\lambda$ is a strongly inaccessible cardinal, then every $\lambda$-tree has a cofinal branch if and only if $\lambda \to (\lambda)^2_2$. The external theorem applies with $\lambda := \kappa$ because the present statement assumes that $\kappa$ is strongly inaccessible, meaning uncountable, regular, and a strong limit cardinal. Its first conclusion is exactly condition 1, since condition 1 defines the tree property as the assertion that every $\kappa$-tree has a cofinal branch. Its second conclusion is exactly condition 2, since condition 2 defines $\kappa \to (\kappa)^2_2$ as the assertion that every map $c:[\kappa]^2 \to 2$ has a homogeneous subset $H \subseteq \kappa$ of cardinality $\kappa$. Therefore condition 1 is equivalent to condition 2. [guided] We need a result that converts the tree formulation into the coloring formulation. The precise external result used here is the Erdős-Tarski combinatorial characterization of weak compactness: for every strongly inaccessible cardinal $\lambda$, the following two assertions are equivalent. 1. Every $\lambda$-tree has a cofinal branch. 2. The partition relation $\lambda \to (\lambda)^2_2$ holds. We now verify that this theorem applies to the present cardinal. The theorem requires $\lambda$ to be strongly inaccessible. In the formalized statement, $\kappa$ is assumed to be strongly inaccessible, namely uncountable, regular, and a strong limit cardinal. Thus we may substitute $\lambda := \kappa$ in the Erdős-Tarski characterization. After this substitution, the first assertion of the external theorem says that every $\kappa$-tree has a cofinal branch. This is exactly condition 1, because the formalized statement explicitly defines the tree property in that way. The second assertion says $\kappa \to (\kappa)^2_2$. This is exactly condition 2, because the formalized statement expands the partition relation as follows: for every coloring $c:[\kappa]^2 \to 2$, there is a set $H \subseteq \kappa$ with $|H|=\kappa$ such that $c$ is constant on $[H]^2$. Therefore the Erdős-Tarski characterization proves precisely that condition 1 holds if and only if condition 2 holds. [/guided] [/step] [step:Apply the Keisler-Makkai compactness characterization to compare the tree property and $L_{\kappa,\kappa}$ compactness] We invoke the Keisler-Makkai model-theoretic characterization of weak compactness: if $\lambda$ is a strongly inaccessible cardinal, then $\lambda$ has the tree property if and only if $L_{\lambda,\lambda}$ satisfies $\lambda$-compactness for every language $\mathcal{L}$ with $|\mathcal{L}| \leq \lambda$ and every $L_{\lambda,\lambda}(\mathcal{L})$-theory $T$ with $|T| \leq \lambda$, where $\lambda$-compactness means that $T$ has a model whenever every subtheory $T_0 \subset T$ with $|T_0|<\lambda$ has a model. The external theorem applies with $\lambda := \kappa$ because $\kappa$ is strongly inaccessible by hypothesis. Its tree-property side is condition 1. Its compactness side has the same language-size bound, $|\mathcal{L}| \leq \kappa$, the same theory-size bound, $|T| \leq \kappa$, the same small-subtheory requirement, $|T_0|<\kappa$, and the same conclusion that $T$ has a model. Thus condition 1 is equivalent to condition 3. [/step] [step:Combine the two characterizations] From the combinatorial characterization, condition 1 is equivalent to condition 2. From the model-theoretic characterization, condition 1 is equivalent to condition 3. Therefore all three conditions are equivalent. Explicitly, if condition 1 holds, then both condition 2 and condition 3 hold by the two cited characterizations. If condition 2 holds, then condition 1 holds by the combinatorial characterization, and hence condition 3 holds by the model-theoretic characterization. If condition 3 holds, then condition 1 holds by the model-theoretic characterization, and hence condition 2 holds by the combinatorial characterization. This proves the theorem. [/step]