On Erdős-Sós Conjecture for Trees of Large Size

Erdős and Sós conjectured that every graph $G$ of average degree greater than $k-1$ contains every tree of size $k$. Several results based upon the number of vertices in $G$ have been proved including the special cases where $G$ has exactly $k+1$ vertices (Zhou), $k+2$ vertices (Slater, Teo and Yap), $k+3$ vertices (Woźniak) and $k+4$ vertices (Tiner). We further explore this direction. Given an arbitrary integer $c\geq 1$, we prove Erdős-Sós conjecture in the case when $G$ has $k+c$ vertices provided that $k\geq k_0(c)$ (here $k_0(c)=c^{12}{\rm polylog}(c)$). We also derive a corollary related to the Tree Packing Conjecture.