  Preservation theorems provide a direct correspondence between the syntactic
structure of first-order sentences and the closure properties of their
respective classes of models. A line of work has explored preservation theorems
relativised to combinatorially tame classes of sparse structures [Atserias et
al., JACM 2006; Atserias et al., SiCOMP 2008; Dawar, JCSS 2010; Dawar and
Eleftheriadis, 2024]. In this article we initiate the study of preservation
theorems for dense graph classes. In contrast to the sparse setting, we show
that extension preservation fails on most natural dense classes of low
complexity. Nonetheless, we isolate a technical condition which is sufficient
for extension preservation to hold, providing a dense analogue to a result of
[Atserias et al., SiCOMP 2008].
