The Gy\'arf\'as tree packing conjecture states that any set of $n-1$ trees $T_{ 1},T_{ 2},..., T_{ n-1}$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$. We show that $t=1/10n^{ 1/4}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_{ n+1}$ (for $n$ large enough). We also prove that any set of $t=1/10n^{ 1/4}$ trees $T_1,T_2,..., T_t$ such that no tree is a star and $T_i$ has $n-i+1$ vertices pack into $K_{ n}$ (for $n$ large enough). Finally, we prove that $t=1/4n^{ 1/3}$ trees $T_1,T_2,..., T_t$ such that $T_i$ has $n-i+1$ vertices pack into $K_n$ as long as each tree has maximum degree at least $2n^{ 2/3}$ (for $n$ large enough). One of the main tools used in the paper is the famous spanning tree embedding theorem of Koml\'os, S\'ark\"ozy and Szemer\'edi.
Categories: math.CO