On the tree packing conjecture

The Gyárfás tree packing conjecture states that any set of n-1 trees T₁, T₂, Tn-1 such that Ti has n-i+1 vertices packs into K_n (for n large enough). We show that t = 1 10n1/4 trees T₁, T₂,Tt such that Ti has n-i+1 vertices packs into K_n+1 (for n large enough). We also prove that any set of t = 1 10n1/4 trees T₁, T₂, Tt such that no tree is a star and Ti has n-i+1 vertices packs into Kn (for n large enough). Finally, we prove that t = 1 4n1/3 trees T₁, T₂, Tt such that Ti has n - i + 1 vertices packs into Kn 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 Komlós, Sárközy, and Szemerédi [Combin. Probab. Comput., 10 (2001), pp. 397-416].