The Base Run format (at least in this context) involves each game containing up to six teams, with two teams playing at each base, each playing solely against the other team at that base. It's usually played as a triples format. Fixture-wise, it's a bit like Lord of the Ring with only two teams per ring instead of three. Below are some sample round robin fixtures for this, which assume you have three suitable bases.
Team 1 R Team 2 R
Team 1 RR. Team 2 R.R Team 3 .RR
Team 1 RRB Team 2 RBG Team 3 GBB Team 4 GRG
Team 1 R.GBB Team 2 RGRG. Team 3 GG.BR Team 4 GBR.B Team 5 .BGGR
Team 1 RBBRR Team 2 RRGGG Team 3 GRBBB Team 4 GGRGR Team 5 BBRBG Team 6 BGGRB
Team 1 RG.GRBB Team 2 BRG.GRB Team 3 BBRG.GR Team 4 RBBRG.G Team 5 GRBBRG. Team 6 .GRBBRG Team 7 G.GRBBR
Team 1 R.G.GRBBR. Team 2 BR.G.GRBB. Team 3 BBR.G.GRG. Team 4 RBBR.G.G.R Team 5 GRBBR.G.R. Team 6 .GRBBR.GB. Team 7 G.GRBBR.G. Team 8 .G.GRBBR.R
Team 1 R..G.GRBBRG. Team 2 BR..G.GRBB.R Team 3 BBR..G.GRG.B Team 4 RBBR..G.G.RG Team 5 GRBBR..G.RB. Team 6 .GRBBR..GBG. Team 7 G.GRBBR..G.R Team 8 .G.GRBBR..RB Team 9 ..G.GRBBR.BG
Team 1 R...G.GRBBR.BG. Team 2 BR...G.GRB.R.BG Team 3 BBR...G.GRB.R.B Team 4 RBBR...G.GGB.R. Team 5 GRBBR...G..GB.R Team 6 .GRBBR...GR.GB. Team 7 G.GRBBR....R.GB Team 8 .G.GRBBR..G.R.G Team 9 ..G.GRBBR.BG.R. Team 10 ...G.GRBBR.BG.R
Team 1 R....G.GRBBR.BG.G.. Team 2 BR....G.GRBB.G.R.R. Team 3 BBR....G.GRG..RB.B. Team 4 RBBR....G.G.R.BG.G. Team 5 GRBBR....G.RB...R.R Team 6 .GRBBR....GBG.G.B.. Team 7 G.GRBBR....G.R.RG.. Team 8 .G.GRBBR....RB.B.R. Team 9 ..G.GRBBR...BG.G.B. Team 10 ...G.GRBBR..G.R.RG. Team 11 ....G.GRBBR..RB.B.R
If your tournament contains 2 to 11 teams, simply use the appropriate round robin from above.
The number of games required is n × (n-1) / 6. This means that as the number of teams grows, the number of games for a full round robin increases very quickly. So for 12 or more teams, break teams into pools. (Ideally each pool will contain teams of approximately the same difficulty — you don't want a "hard" pool and an "easy" pool; you want every pool to be "medium".) For 12 teams, use 2 pools of 6; for 13, a pool of 6 and a pool of 7; for 14, 7 and 7; for 15, 7 and 8; for 16, 8 and 8; for 17, 8 and 9. For 18, three pools of 6; and so on. Interleave games from different pools: if you have two pools, this means that every team has a break between every game they play. Note that where pool sizes are uneven, teams in the smaller pool will get less games.
If you feel like your pools are well seeded, you should take a similar number of teams from each pool through to the next round. If they are seeded randomly, you should consider a different advancement strategy which tries to take the difficulty of the different pools into account.
For large tournaments you might then have a round 2 / semi-final of the top 6 teams. (I say 6 because it's the smallest pool that uses all three bases effectively.)
You can then go to a grand final. This might feature the top 2 teams, or the top 4, or the top 6. Note that if the teams in the grand final have all come there from a single pool (whether that pool was round 1 or round 2), then those teams will have necessarily played each other — if the grand final is just 2 teams, you can make the final outcome the average of the grand final game and the other games they played each other in. If they've played each other on two of the three colours already, you may wish to make the grand final play at the third colour.
If you plan to use a round 1, round 2, then finals, the ideal number of teams to have in round 2 is √(number of teams in round 1) × (number of teams in finals). This gives the same "cut rate" from round 1 to round 2 as from round 2 to finals. You'll need to round to an integer, obviously, and you should probably round up to 6 if necessary, in order to make effective use of all the bases.
(Here is an explanation of how to read a grid.)