Social Golfer Problem - Quintets
Solution 1:
Have you seen Ivan Dotu and Pascal Van Hentenryck, Scheduling social golfers locally? They claim
45 golfers in groups of 5 can play 7 days;
54 golfers in groups of 6 can play 6 days;
50 golfers in groups of 5 can play 8 days;
and some other results that lie outside the bounds of your question. But they don't actually present any instances of the solutions, just their algorithm and the amount of time it took to find the instances.
Solution 2:
I don't have a lot of advice on quintets in particular, but perhaps the information below will be useful nevertheless.
Some of the solutions you are looking for correspond to the necessary conditions for a resolvable balanced incomplete block designs and, if such a RBIBD exists, then this will lead immediately to a solution with the maximal number of days. The items in this category are:
- 33 play in groups of 3 for 16 days - the much studied Kirkman Triple System
- 40 play in groups of 4 for 13 days - see Kageyama (1972)
- 52 play in groups of 4 for 17 days - see Lam & Miao (1999) Lemma 5.3
- 25 play in groups of 5 for 6 days - see comment above
- 45 play in groups of 5 for <=11 days - RBIBD is an open problem
- 36 play in groups of 6 for <7 days - RBIBD does not exist
- 66 play in groups of 6 for <=13 days - RBIBD is an open problem
In addition to the above, I can offer examples of the following:
- 30 play in groups of 3 for 14 days [maximal]
- 36 play in groups of 3 for 17 days [maximal]
- 44 play in groups of 4 for 13 days
- 48 play in groups of 4 for 14 days [see Ge and Lam(2003) Lemma 3.1]
- 45 play in groups of 5 for 9 days.
I have also been confused by the large literature on this subject. I feel that 44/4/14 days, and 48/4/15 days should both be possible, but I have not come across a construction yet.
References:
Ge & Lam (2003) "Resolvable group divisible designs with block size four and group size six"; Discrete Mathematics, 268, 139 – 151.
Kageyama (1972) "A survey of resolvable solutions of balanced incomplete block designs"; Rev. Inst. Internat. Statist., 40, 269–273.
Lam & Miao (1999) "On Cyclically Resolvable Cyclic Steiner 2-Designs", Journal of Combinatorial Theory, Series A 85, 194-207.