  We study Nash equilibria in the network creation game of Fabrikant et
al.[10]. In this game a vertex can buy an edge to another vertex for a cost of
$\alpha$, and the objective of each vertex is to minimize the sum of the costs
of the edges it purchases plus the sum of the distances to every other vertex
in the resultant network. A long-standing conjecture states that if $\alpha\ge
n$ then every Nash equilibrium in the game is a spanning tree. We prove the
conjecture holds for any $\alpha>3n-3$.
