Need math help with axiomatic systems and theorems

Axiom 1: Each game is played by two distinct teams 

Axiom 2: There are at least four teams 

Axiom 3: There are at least six games played 

Axiom 4: Each team played at most 4 games. 

I need help developing a third theorem and verifying the second is true. 

Theorem 1: If there are exactly 4 teams, then there are at most 8 games. 

Theorem 2: If there are exactly 6 games, then there are at most 12 teams. 

Theorem 3: