  We introduce $ \Lambda(\Phi) $-sets as generalizations of $ \Lambda(p)
$-sets. These sets are defined in terms of Orlicz norms. We consider
$\Lambda(\Phi)$-sets when the Matuszewska-Orlicz index of $ \Phi $ is larger
than $ 2 $. When $S$ is a $\Lambda(\Phi)$-set, we establish an estimate of the
size of $ S \cap [-N,N] $ where $ N \in \mathbb{N} $. Next, we construct a $
\Lambda(\Phi_1)$-set which is not a $ \Lambda(\Phi_2)$-set for any $ \Phi_2 $
such that $ \sup_{u \geq 1} \Phi_2(u) / \Phi_1(u) = \infty $ by using a
probabilistic method. With an additional assumption about a subset $E$ of
$\mathbb{Z}$, we can construct such a $\Lambda(\Phi_1)$-set contained in $E$.
These statements extend known results on the structure of $ \Lambda(p) $-sets
to $\Lambda(\Phi)$-sets.
