I gave a contributed talk at the British Logic Colloquium at the University of Sussex on September 7.
Notes and slides can be found on my talks page. Here’s an abstract:
Does every structure have a proper elementary substructure of the same size? This question — whose answer is false — led to the notion of a Jónsson cardinal, being the cardinals for which this is true. These cardinals turn out to exhibit strange behaviour compared to other large cardinals, as the theory “ZFC + there is a Jónsson cardinal” is quite strong, but a Jónsson cardinal itself is ostensibly very weak in terms of direct implication. I will talk a bit about this asymmetry in strength and zoom in on the problem of determining the direct implication strength, leading to a special relationship between Jónsson cardinals and weakly compact cardinals.