  Recently, Balogh--Morris--Samotij and Saxton--Thomason proved that
hypergraphs satisfying some natural conditions have only few independent sets.
Their main results already have several applications. However, the methods of
proving these theorems are even more far reaching. The general idea is to
describe some family of events, whose cardinality a priori could be large, only
with a few certificates. Here, we show some applications of the methods,
including counting $C_4$-free graphs, considering the size of a maximum
$C_4$-free subgraph of a random graph and counting metric spaces with a given
number of points. Additionally, we discuss some connections with the
Szemer\'edi Regularity Lemma.
