I have set of participants
$$N = \{1,2,...n \}$$
a set of rounds
$$R=\{1,2,...r\}$$
and a set of teams
$$T=\{1,2,...t\}$$
and need to define a function, $F:(N,R)\to T$ that assigns participants to a team for each round such that the number of instances of team-mates repeating, i.e.,
$$\sum_{r_1,r_2,n_1,n_2}[F(n_1,r_1)=F(n_2,r_1)∧F(n_1,r_2)=F(n_2,r_2)]$$
is minimised. In particular, for a given $N, R, T$ is a solution possible where this sum is zero?
For computational complexity I am trying to solve for $n=22, r=4, t=4.$
The problem can be formulated as either a mixed-integer linear program (MILP) or a constraint programming (CP) problem. Theoretically, a good quality solver of either type could solve it, provided that you were able to come up with a good search strategy for the CP solver or a good formulation (and maybe some clever solver parameter choices) for the MILP.
I took a whack at the MILP approach. One major stumbling block is that the problem has a lot of symmetry. The symmetry can be mitigated to some extent, but the combination of that and maybe a somewhat loose formulation (at least the way I did it) make for rather slow solver progress. Assuming I didn't screw anything up (a very charitable assumption), it appears that you cannot get by without at least 14 pair repetitions, and should be able to do no worse than 23.
If you are willing to settle for a good solution (not a provably optimal one), a heuristic or metaheuristic might work well for you.