The Covering Reflection Property holds at a cardinal \(\kappa\) if for every first order structure \(\mathcal B\) in a countable language, there is some \(\mathcal A\) of size \(<\kappa\) so that \(\mathcal B\) can be covered with the ranges of elementary embeddings \(j:\mathcal A\rightarrow \mathcal B\). That is, for every \(b\in\mathcal B\), there is some \(a\in\mathcal A\) and an elementary embedding \(j:\mathcal A\rightarrow\mathcal B\) with \(j(a)=b\). We discuss this property and isolate a new large cardinal notion strictly between almost huge and huge cardinals and show that the least cardinal exhibiting the Covering Reflection Property is exactly the least such large cardinal. Moreover, there is a natural correspondence between such large cardinals and strong forms of the Covering Reflection Property.

This is joint work with Joel D. Hamkins, Nai-Chung Hou and Farmer Schlutzenberg.