Let's say there are 10 golfers playing a mini 3 round tournament, 1 round on each of 3 consecutive days. Each day they are split in to 2 groups of 5 golfers. Is it possible that, by the time the tournament is over, each golfer has played with each other golfer at least once?
Follow up question: If it is not possible, what is the largest grouping size for which it is possible?
Edit: By "largest grouping size" I mean, what is the largest N such that N golfers can be split in to 2 groups of N/2 each day and by the end each of the N golfers will have played with each other golfer.