Positional games with efficient winning strategy

We study games where two players are coloring edges of infinite complete graph. Both players are trying to create given target subgraph colored by their colors. Firstly, we will look at situation when the target subgraph is a complete graph. We will show that first player has winning strategy if the target subgraph is complete graph on at most three vertices. Then we will slightly change the rules. This will help us show that the first player has a winning strategy if he can bound the size of the complete graph on which the game is played or he can color a few edges more then the opponent. At the end we will discuss game, when a complete graph of a given size is a minor of target subgraph. We will show a winning strategy for the first player and for small size of complete graph which is the minor of target subgraph. We also will discuss, why the first player should have winning strategy for all games of this type.