Creating Groups, Minimal Overlap

[gumpzilla]

No, it isn't possible with four games. An easy way to see this is graphical. Consider six points A, B, C, D, E and F. When A, B and C play a game together, I'll draw a triangle connecting them. Your requirement that every team play every team at least once means that when I've put in all the triangles I'm going to put in, there is an edge connecting every two points. This requires 6*5/2 = 15 edges. But with four triangles I only have 12 edges. So it's impossible to have every player play a game with every other player at least once with four games.