Here is a round robin as I know it:
https://www.aceodds.com/what-is-a-round-robin-bet.html
http://www.tiptime.co.uk/horse-racing-multiple-bets/
What you have described is an each way Yankee:
4 horses consists of 6 doubles;4 trebles and one accumulator (4 fold here).
Equals 11 bets or 22 bets for w/p.
5 horses consists of 10 doubles;10 trebles;5 four folds and one accumulator.
6 horses is a Heinz 57 since there are 57 bets
15 doubles;20 trebles;15 four folds;6 five folds and one accumulator.
You can make these bets yourself;picking horses at different tracks;but you
would need to be able to track going on money and indeed you would adversely
effect your own outcome by money running on at the track.
Having them in betting pools would be different;but can you realistically see
that happening today?
Can you believe that I used to work these out manually;hundreds of times per
day;six days per week in Belfast and in London during the early sixties.
