A Sidon set has all pairwise sums distinct. The open questions concern the largest Sidon set in {1, …, n}, infinite Sidon sequences, and variants with more summands. Many reduce to constructions or computations that can be verified exactly.
Find the shortest Golomb ruler (all pairwise mark differences distinct) with n marks. Optimality is proven up to 28 marks (length 585, distributed.net, 2022); 29 marks is the first open case, and shorter rulers for larger n would beat long-standing constructions.
Does 1,2^3,…,N^3 contain a Sidon set of size ≫ N?
Let A ⊆ ℕ. Let B ⊆ ℕ be the set of integers which are representable in exactly one way as the sum of two elements from A. Is it true that for all ε > 0 and large N, |1,…,N ∖ B| ≫_ε N^1/2 - ε?
Let A be a finite Sidon set and A+A=s_1<⋯<s_t. Is it true that 1/tΣ_1≤ i<t(s_i+1-s_i)^2 → ∞ as lvert Arvert→ ∞?
Is it true that for every k ≥ 1 we have F(N + k) ≤ F(N) + 1 for all sufficiently large N?
Does there exist a maximal Sidon set A⊂ 1,…,N of size O(N^1/3)? A question of Erdős, Sárközy, and Sós [ESS94].
Let A be an infinite B₂[2] set. Must liminf |A ∩ 1, ..., N| * N ^ (- 1 / 2) = 0?
Is it true that f(N)∼ N^1/3? Originally asked to Erdős by Bose. This is discussed in problem C11 of Guy's collection [Gu04].
Is it true that, for every ε > 0, h(N) = sqrt N + O_ε(N^ε)
Does there exist a polynomial f(x)∈ℤ[x] such that all the sums f(a)+f(b) with a < b nonnegative integers are distinct?
Erdős Problem 329. Let A ⊆ ℕ be a Sidon set. How large can lim sup_N → ∞ |A ∩ 1,…,N| / N^1/2 be?
Let A = 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, … be the greedy Sidon sequence: we begin with 1 and iteratively include the next smallest integer that preserves the Sidon property (i.e. there are no non-trivial solutions to a + b = c + d). What is the order of growth of A?
Is there an infinite Sidon set A⊂ ℕ such that lvert A∩ 1…,Nrvert ≫_ε N^1/2-ε for all ε > 0?
Let A ⊂ ℕ be an infinite set such that the triple sums a+b+c are all distinct for a,b,c ∈ A (aside from the trivial coincidences). Is it true that liminf_N → ∞ fraclvert A ∩ 1,…,NrvertN^1/3=0?
Erdős Problem 44: Let N ≥ 1 and A ⊆ 1,…,N be a Sidon set. Is it true that, for any ε > 0, there exist M = M(ε) and B ⊆ N+1,…,M such that A ∪ B ⊆ 1,…,M is a Sidon set of size at least (1−ε)M^1/2?
What is the supremum of the set of admissible numbers?
What is the size of the largest Sidon subset A⊆1,2^2,…,N^2? Is it N^1-o(1)?
