Let C ⊂≠ Reg be a non-empty class (of regular cardinals). Then the logic has additional nice properties: it has the homogeneous model existence property.
@article{1371,
title = {On $\lambda$ strong homogeneity existence for cofinality logic},
journal = {CUBO, A Mathematical Journal},
volume = {13},
year = {2011},
language = {en},
url = {http://dml.mathdoc.fr/item/1371}
}
Shelah, Saharon. On λ strong homogeneity existence for cofinality logic. CUBO, A Mathematical Journal, Tome 13 (2011) . http://gdmltest.u-ga.fr/item/1371/