Does there exist a supercompactness theorem? Large cardinals such as weakly compact cardinals, measurable cardinals, strongly compact cardinals, and extendible cardinals all can be characterized in terms of a certain compactness theorem of certain strength.
For instance, weakly compact cardinals are precisely the cardinals $\kappa$ where the $\kappa$-compactness theorem holds in $L_{\kappa,\kappa}$ when we bound the number of non-logical symbols by $\kappa$. Strongly compact cardinals are precisely the cardinals where the $\kappa$-compactness theorem holds for $L_{\kappa,\kappa}$. Extendible cardinals are the cardinals where the compactness theorem holds for higher order infinitary logic. Furthermore, there are characterizations of at least weakly compact, measurable, and strongly compact cardinals involving different forms of compactness besides logical compactness.
Is there is a characterization of supercompact cardinals in terms of some sort of compactness theorem? Can any other large cardinal axioms besides the ones already mentioned be characterized in terms of some sort of compactness theorem?
 A: Now I see you are also asking for combinatorial forms of compactness. I am not sure whether the tree property should be considered as a form of "compactness", but there are results by Jech, Di Prisco, Magidor, Zwicker and perhaps others which give characterizations of large cardinals in terms of variations of the tree property, and recent interest in such matters (together with new results!) has originated with Christoph Weiß'  Dissertation of 2010, Subtle and Ineffable Tree Properties.
Similar to the case of the classical tree property, such stronger  tree properties characterize large cardinals only under the assumption of (strong) inaccessibility. Without inaccessibility these tree properties can hold on small cardinals, too.
Besides Weiß'  Dissertation, further results of this kind appear, for example,  in Weiß, The combinatorial essence of supercompactness, Ann. Pure Appl. Logic 163 (2012), Viale and Weiß, On the consistency strength of the proper forcing axiom, Adv. Math. 228 (2011), no. 5, 2672-2687,  Fontanella, Strong tree properties for two successive cardinals, Arch. Math. Logic 51 (2012), no. 5-6, 601-620. References to former results can be found in the above works, and also in the MR review  of Laura's paper, MR2945570 by Andrew D. Brooke-Taylor.
At present nothing similar comes to my mind for larger cardinals, say, huge ones. 
A: The answer is yes. In the paper "Compactness for omitting of types" (Annals of Mathematical Logic, vol 14(1) (1978), pp. 39-56), Benda gives such a characterization. He proves the following:
Theorem. The following are equivalent:
1) $\kappa$ is supercompact,
2) If $T$ is a theory in $L_{\kappa, \kappa}$ and $\Sigma(x,y)$ is a type such that $\{ \Delta \in P_{\kappa}(\Sigma) : T+(\exists x)\alpha_\Delta(x)$ has a model$ \}$ is in $ F_{\kappa}(\Sigma),$ then $T+(\exists x)\alpha_\Sigma(x)$ has a model.
Here  $\alpha_\Delta(x)$ is the formula $\bigwedge(\exists y)\Delta \wedge (\exists y)\bigwedge\Delta $ and $F_{\kappa}(\Sigma)$ is the filter of club subsets of $P_{\kappa}(\Sigma)$
