CMO 2000 Problem 3

A table tennis club hosts a series of doubles matches following several rules: (i) each player belongs to two pairs at most; (ii) every two distinct pairs play one game against each other at most; (iii) players in the same pair do not play against each other when they pair with others respectively. Every player plays a certain number of games in this series. All these distinct numbers make up a set called the "set of games". Consider a set $A=\{a_1,a_2,\ldots ,a_k\}$ of positive integers such that every element in $A$ is divisible by $6$ . Determine the minimum number of players needed to participate in this series so that a schedule for which the corresponding set of games is equal to set $A$ exists.