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Updated Oct 7

Problem 773

Statement. What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$ ? Is it $N^{1-o(1)}$ ?

Status. Open.

Source. erdosproblems.com/773, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #773, https://www.erdosproblems.com/773.

References.

[AlEr85] Alon, Noga and Erdős, P., An application of graph theory to additive number theory. European J. Combin. (1985), 201-203.
[Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
[LeTh95] Lefmann, Hanno and Thiele, Torsten, Point sets with distinct distances. Combinatorica (1995), 379-408.

Formalization. Statement in formal-conjectures.

Progress

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Known Results

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Linked library material

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Linked from (11)

Additive Bases and Sidon Sets An application of graph theory to additive number theory Remark (p. 203): the squares up to n^2 contain a Sidon subset of c(eps) n^{2/3-eps} terms and none of more than c' n/(log n)^{1/4}additive_bases/bosio_2026_large_b_2_g_subsets_first Theorem 1.1 (p. 2): large B_2[g] subsets of the first n squares additive_bases/croot_2026_combinatorial_large_sieve_sidon_sets_distances Corollary 1.2 (p. 3): Sidon subsets of the first N squares Theorem 1.1 (p. 2): Sidon sets in [N] missing a fixed fraction of residues mod every prime additive_bases/garaev_2026_sidon_sets_squares_cubes_quartics_short Theorem 1 (pp. 2--3): the squares n^2 with N <= n < N + (8N+8)^(1/2) + 2 form a Sidon set Theorem 2 (p. 3): the squares n^2 with N <= n <= N + ((8+eps)N)^(1/2) are not a Sidon set for large N

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