Advances on the Conjecture of Erd\H{o}s-S\'os for spiders

- A hamiltonian graph $G$ verifying $e(G)>n(k-1)/2$ %with a vertex of degree greater or equal than $k$ contains any $k$-spider. - If $G$ is a graph with average degree $\bar{d} > k-1$, then every spider of size $k$ is contained in $G$ for $k\le 10$. - A $2$-connected graph with average degree $\bar{d} > \ell_2+\ell_3+\ell_4$ contains every spider of $4$ legs $S_{1,\ell_2,\ell_3,\ell_4}$. We claim also that the condition of $2$-connection is not needed, but the proof is very long and it is not included in this document.