2015 Countdown Problem #17: A Graph Theory Problem

Suppose there are 2015 vertices. Let a complete graph on $n$ vertices be denoted as $K_n$ .

Which statements are true?

I. $K_{2015}$ has 2029105 edges.

II. Suppose two $K_{2015}$ graphs are connected together, by having exactly one edge connected between any vertex in each of the original $K_{2015}$ graphs, to form a new graph $G_{2015}$ . Then $G_{2015}$ is bipartite.

III. Let $H_{2014}$ be a complete 2014-partite graph on $2014^2$ vertices, such that the number of vertices in each partition are equal. Then $K_{2015}$ is not a subgraph of $H_{2014}$ .

This problem is part of the set 2015 Countdown Problems .