  We study four different models of Chern insulators in the presence of strong
electronic repulsion at partial fillings. We observe that all cases exhibit a
Laughlin-like phase at filling fraction 1/3. We provide evidence of such a
strongly correlated topological phase by studying both the energy and the
entanglement spectra. In order to identify the key ingredients of the emergence
of Laughlin physics in these systems, we show how they are affected when tuning
the band structure. We also address the question of the relevance of the Berry
curvature flatness in this problem. Using three-body interactions, we show that
some models can also host a topological phase reminiscent of the nu=1/2
Pfaffian Moore-Read state. Additionally, we identify the structures indicating
cluster correlations in the entanglement spectra.
