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Problem 44 — Erdős Problems

Problem 44

Let $N\geq 1$ and $A\subset \{1,\ldots,N\}$ be a Sidon set. Is it true that, for any $\epsilon>0$ , there exist $M$ and $B\subset \{N+1,\ldots,M\}$ (which may depend on $N,A,\epsilon$ ) such that $A\cup B\subset \{1,\ldots,M\}$ is a Sidon set of size at least $(1-\epsilon)M^{1/2}$ ?

Source: erdosproblems.com/44

Number theory
Sidon sets
Additive combinatorics

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Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; section C9 "Packing sums of pairs", printed p. 177: whether a finite Sidon sequence can be prolonged to one with $a_n<(1+o(1))n^2$ , "as dense as possible asymptotically". Library home: Guy 2004.

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Status: Open
Topics: Number theory
Sidon sets
Additive combinatorics
In the wiki: Wiki page