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# Synonyms...

I need some words, some adjectives. Pretend for a moment that you are a coordinator for social events like speed dating. Your event will have some number of rounds. During each round, you will pair up the participants.

I need some adjectives to describe events. For example, you might call an event complete if every potentially compatible pair got paired up in some round of the event. I would prefer words without negating prefixes when possible.

So, I need adjectives for the following concepts:

1. Every potentially compatible pair was paired during some round. (complete, covered, total, ???)
2. No one was ever paired with the same person in more than one round. (nonredundant, unrepeating, drive-by, passing, ???)
3. Say that n is the maximum number of people that any participant is potentially compatible with, and we manage to get together every compatible pair in only n rounds. (speedy, efficient, fast, ???)
4. No one ever has to sit out for a round. (engaging, efficient, non-idle, busy, ???)

Let P be the set of participants. Let C be a subset of the power set of P. Call the elements of C the potentially compatible groupings. For a typical speed-dating scenario, all potentially compatible groupings would be of size-two (and, for a heterosexual event would consist of one male and one female in each), but we're going to allow for more general groupings.

Let S be a set of rounds { R1, R2, ..., Rk } where each round is a subset of the power set of C and we have ∩ { g ∈ Rj } = ∅ for all j. (The gist here is that no person can participate in two different groups in a single round.)

The above-mentioned situtations that I want adjectives for are more formally:

1. ∀ g ∈ C there exists Rj ∈ S with g ∈ Rj.
2. g ∈ Ri and g ∈ Rj if and only if i = j.
3. Condition #1 is true and | S | = maxx ∈ P | { g ∈ C | x &isin g } |
4. ∩ { &cup Rj | Rj ∈ S } = P.

There are obviously a ton of combinatoric questions from here. For example, given a simple graph G, let C be the set of all pairs of vertexes which share an edge. Which of the above adjectives can be realized for G = Km,n or G = Kn? Or, how many distinct S are there for a given P and C?

I wouldn't be surprised at all to find there's already a body of mathematical literature most or all of this. But, I'm not sure at all what to search for. Any pointers would be much appreciated.

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