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I'm trying to understand Vopěnka's Principle, which is a large cardinal axiom. One version of the principle is that there does not exist a proper class of directed graphs such that there are no homomorphisms between any two graphs in the class. This is a large cardinal axiom because it implies the existence of a proper class of measurable cardinals. If Vopěnka's principle is true, then it is not provable in ZFC, since it implies the consistency of ZFC. (It's falsity may be provable in ZFC.)

I presume that since it functions as a large cardinal axiom, then it must fail at small cardinals (i.e. cardinals whose existence is provable within ZFC, such as $\aleph_\omega$). Is there an explicit construction for any such cardinal $\kappa$ there exists a set of graphs of size $\kappa$ such that Vopěnka's Principle fails for that set? In other words, there are no two homomorphisms between the two graphs in that set?

I can come up with a construction for $\aleph_0$, but that's it. (For $\aleph_0$, the set of directed cycle graphs with a prime number of vertices does the trick, I think.)

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2 Answers 2

up vote 5 down vote accepted

If $\kappa$ is almost huge (another large cardinal property), then for each family of size $\kappa$ of graphs of size $<\kappa$, one of the graphs embeds into another one from the family. See this post by Harvey Friedman.

Looking at $\kappa$-many graphs of size $<\kappa$ seems to be the right set analog to a class of graphs (that are sets).

Could you clarify whether your family is of size $\kappa$, or the graphs?


Added after arsmath's comment: If you scroll down in Friedman's note, he says that if $\kappa$ is Vopenka, then the set of extendible cardinals below $\kappa$ is stationary in $\kappa$. Extendibility implies measurability (this should be in Kanamori's "The Higher Infinite") and hence there is a measurable cardinal below every Vopenka cardinal.
I think this qualifies as "no small cardinal is Vopenka".

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The family is of size $\kappa$. I guess what I'm wondering is if you can prove in ZFC that small, everyday cardinals (I don't know what the technical term is) are <i>not</i> Vopěnka cardinals. –  arsmath Sep 23 '10 at 16:17
    
Thanks! That does answer the original question. I was hoping for explicit construction. I edited the question accordingly. –  arsmath Sep 23 '10 at 17:19

I found a reference for an explicit construction, which I am recording here for posterity: Chapters 2G and 6A of Locally Presentable and Accessible Categories by Ademek and Rosicky.

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