On The Erd(?)s-Sós Conjecture

Erd（?）s-Sós conjecture（1962）is that if G is a graph with n vertices and the number of edges of G is more than k-1,then G contains all trees with k+1 vertices.Researchers have studied the conjecture under various additional conditions,but the conjecture has not been resolved completely.This thesis collects results of predecessors and mainly considers the case that the number of vertices of T is closed to that of G.We improve Agnieszka G?rlich and Andrzej?ak’s result.Their result is that G is a graph with n=k+cG contains all trees with k+1 vertices.We improve the power of c to 9+ε:G is a graph with n=k+c vertices and k≥k₀（c）(k₀（c）≥γc^9+εwhereγis a sufficiently large constant,ε>0),then G contains all trees with k+1 vertices.