The aim of the talk is to present a proof of the
Enflo-Rosenthal theorem stating that Banach spaces of the form $L_p(\mu)$
(where $p\in (1,+\infty)$, $p\ne2$ and $\mu$ is a finite measure) of the density
dens$(L_p(\mu))\ge \kappa_\omega$ do not have unconditional bases. A combinatioral lemma
similar to the Kuratowski's free set theorem is a crucial step in the
proof. During the talk we will prove both of the aforementioned
combinatorial results and proceed to the proof of the Enflo-Rosenthal
theorem. The problem whether the assumption on the density of the space
can be weakened to dens$(L_p(\mu))\ge \omega_1$ seems to be still open.