MHB Creating theorems for an axiomatic system

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The discussion centers on formulating theorems based on a defined set of axioms regarding team games. The established theorem states that with exactly four teams, there can be at most eight games. Participants are encouraged to explore additional theorems by fixing one variable and analyzing the implications for the others. Key questions include determining the maximum number of teams possible with exactly six games played and the scenario where each team plays only once. The conversation emphasizes the importance of understanding the relationships between the number of teams, games, and game participation limits.
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I have the axioms:

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.

And I have come up with the theorem: If there are exactly 4 teams, then there are at most 8 games.

I need to come up with two more theorems. I'm pretty stuck on where to go from here, so any advice would be greatly appreciate
 
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How many teams can there be, if there are exactly six games played?

If each team plays only once, how many teams can there be (if only teams that play games count)?

The general idea is: fix ONE of your variables, and determine limits on the others.
 

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