I believe the model-theoretic version of the question is: is there a theory in finitary first-order logic which has, for each cardinal $\lambda$, a set $C\_\lambda$ of $\lambda$-many models, such that if $M,N \in C\_\lambda$ then there is no elementary embedding from $M$ to $N$ or vice versa (in ZFC)?




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One statement of the large cardinal axiom [Vopěnka's Principle](https://en.wikipedia.org/wiki/Vop%C4%9Bnka%27s_principle) is that no accessible category has a full subcategory which is both large and discrete.


Adámek and Rosický [point out](http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511600579) (Remark 6.2(2)) that for any cardinal $\lambda$, it's trivial to come up (in ZFC) with an accessible category with a full discrete subcategory with $\lambda$-many objects. They use the example of the theory $\mathbf{Rel}\_\lambda$, with $\lambda$-many unary relation symbols, and the set of objects $A\_i$, each carried by the one-point set, where $A\_i$ has just the $i$th relation turned "on". Here the accessible category in question is allowed to vary with the cardinal $\lambda$.


But, in ZFC, is there one single accessible category $\mathcal{K}$ which has a full, discrete subcategory $\mathcal{K}\_\lambda \subset \mathcal{K}$ of cardinality $\lambda$, for each cardinal $\lambda$?


Of course, the union $\cup\_\lambda \mathcal{K}\_\lambda$ is large, so (assuming that Vopěnka's principle is consistent over ZFC), if such a category exists, then one won't be able to show that $\cup\_\lambda \mathcal{K}\_\lambda$ is discrete. But it could be that all of its morphisms go from objects of one $\mathcal{K}\_\lambda$ to another $\mathcal{K}\_{\lambda'}$, and the $\mathcal{K}\_\lambda$'s themselves might all be discrete.


**Bonus question:** in your example, are there clearly morphisms between objects in different $\mathcal{K}\_\lambda$'s, or is your example a candidate to become a counterexample to Vopenka in some models (in which connection, [this question](https://mathoverflow.net/questions/203576/explicit-counter-example-to-vop%C4%9Bnkas-principle-in-the-constructible-universe) may be relevant)?



Around Vopěnka: Accessible category with small full discrete subcategories of arbitrary size?
The answer is yes. One can do this with pointed directed graphs.


Specifically, for any infinite cardinal $\lambda$, let $C\_\lambda$ consist of
all structures of the form $\langle V\_{\lambda+2},{\in},\beta\rangle$,
where $\beta<\lambda$ and $V\_{\lambda+2}$ consists of the sets of von Neumann rank at most $\lambda+1$. So this is a pointed directed graph. Since there are $\lambda$ many choices for the constant $\beta$, we have $\lambda$ many models here. 


But there can be no elementary embedding between any two such
structures, since any such embedding would give rise to a
nontrivial elementary embedding $j:V\_{\lambda+2}\to V\_{\lambda+2}$, which is
impossible by the Kunen inconsistency.


