If I'm giving a lecture on trivial definable closure (as a property of graphs), what is a good example of a graph that I can easily draw to depict the concept of trivial dcl?
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1$\begingroup$ Can you please tell me the definition of definable closure? I would be interested in knowing it, but I am not able to find it quickly. $\endgroup$– Valerio CapraroOct 31, 2011 at 16:37
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$\begingroup$ @ValerioCapraro: There is a thorough treatment in Wilfrid Hodges' books Model Theory and A Shorter Model Theory. Just like the (model-theoretic) algebraic closure of a set consists of all elements realising types (equivalently: formulas) over the set that have only finitely many solutions, the definable closure consists of all elements realising types (equivalently: formulas) over the set that have only one solution. E.g., in algebraically closed fields, the definably closed sets are the subfields, and the algebraically closed sets are the algebraically closed subfields. $\endgroup$– user12283Oct 18, 2015 at 14:27
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The infinite edgeless graph has trivial definable closure, since for any finite tuple $\bar a$, there are automorphisms fixing $\bar a$ and swapping any two vertices outside of $\bar a$.
But of course, the infinite edgeless graph is so not interesting to have drawn on the chalkboard. (I suppose you could address this by drawing the complete graph on infinitely many vertices, which also has trivial definable closure for the same reason.) But I would recommend another example, the countable random graph. For any finite tuple $\bar a$ and any vertex $b$ outside $\bar a$, there is an automorphism fixing $\bar a$ and moving $b$, and so the definable closure of $\bar a$ is itself. This situation would be a good excuse to discuss the random graph, if you have not done so already.
answered Oct 31, 2011 at 7:56
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2$\begingroup$ I really don't know the answer to this, but it seems that the Henson graphs, $H_n$ (Fraisse limits of the class of finite graphs with $K_n$, the complete graph on $n$ vertices banned) should have trivial dcl, but I don't know. On the other hand, in the case of the random graph, the complement is the random graph as well, but with these graphs, their complements are actually different, and they should also have the property, right? Can anyone think of any other examples (besides by doing things like taking disjoint unions of these graphs or finite empty or complete graphs)? $\endgroup$ Oct 31, 2011 at 22:15