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.