Is it true that for all $\epsilon>0$ and large $N$ \[\lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/2-\epsilon}.\] Is it true that \[\lvert \{1,\ldots,N\}\backslash B\rvert =o(N^{1/2})?\]
Is it true that for all $\epsilon>0$ and large $N$ \[\lvert \{1,\ldots,N\}\backslash B\rvert \gg_\epsilon N^{1/2-\epsilon}.\] Is it true that \[\lvert \{1,\ldots,N\}\backslash B\rvert =o(N^{1/2})?\]
Erdös and Freud investigated the finite analogue in 'a recent Hungarian paper', proving that there exists $A\subseteq \{1,\ldots,N\}$ such that the number of integers not representable in exactly one way as the sum of two elements from $A$ is $<2^{3/2}N^{1/2}$, and suggest the constant $2^{3/2}$ is perhaps best possible.
Erdős and Rényi have constructed, for any $\epsilon>0$, a set $A$ such that \[\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}\] for all large $N$ and $1_A\ast 1_A(n)\ll_\epsilon 1$ for all $n$.
More generally, Bose and Chowla conjectured that the maximum size of $A\subseteq \{1,\ldots,N\}$ with all $r$-fold sums distinct (aside from the trivial coincidences) then \[\lvert A\rvert \sim N^{1/r}.\] This is known only for $r=2$ (see [30]).
This sequence is at OEIS A005282.
If $A\subseteq \{1,\ldots,n\}$ is such that all products $a_1\cdots a_r$ are distinct for $a_1<\cdots <a_r$ then is it true that \[\lvert A\rvert \leq \pi(n)+O(n^{\frac{r+1}{2r}})?\]
In particular, is it true that $\ell(N)\sim N^{1/2}$?
In [AlEr85] Alon and Erdős make the stronger conjecture that perhaps $A$ can always be written as the union of at most $(1+o(1))N^{1/2}$ many Sidon sets. (This is easily verified for $A=\{1,\ldots,N\}$ using standard constructions of Sidon sets.)
Erdős and Sós proved that $c\geq 1/2$. Gyárfás and Lehel [GyLe95] proved \[\frac{1}{2}<c<\frac{3}{5}.\] (The example proving the upper bound is the set of the first $n$ Fibonacci numbers.)
Is it true that $H_k(n)/n^{1/2}\to \infty$? Or even $H_k(n) > n^{1/2+c}$ for some constant $c>0$?
The answer is yes, and in fact \[H_k(n) \gg_k n^{2/3},\] proved by Alon and Erdős [AlEr85]. We sketch their proof as follows: take a random subset $A'\subset A$, including each $n\in A'$ with probability $\asymp n^{-1/3}$. The number of non-trivial additive quadruples in $A$ is $\ll n^2$ and hence only $\ll n^{2/3}$ non-trivial additive quadruples remain in $A'$. Since the size of the random subset is $\gg n^{2/3}$, all of the remaining non-trivial additive quadruples can be removed by removing at most $\lvert A'\rvert/2$ (choosing the constants suitably).
The analogous question with $A-A$ in place of $A+A$ is simpler, and there the maximal size is $\sim N^{1/2}$, as proved by Cilleruelo.
While $A(N)$ has not been completely determined, both of these questions are now settled, the first positively and the second negatively. The current best bounds are (for large $N$) \[2^{1.16f(N)}\leq A(N) \leq 2^{6.442f(N)}.\] The lower bound is due to Saxton and Thomason [SaTh15], the upper bound is due to Kohayakawa, Lee, Rödl, and Samotij [KLRS].
See also [862].
Similarly, let $B\subseteq \{1,\ldots,N\}$ be a set of maximal size such that there are at most $r$ solutions to $n=a-b$ for any $n$.
If $\lvert A\rvert\sim c_rN^{1/2}$ as $N\to \infty$ and $\lvert B\rvert \sim c_r'N^{1/2}$ as $N\to \infty$ then is it true that $c_r\neq c_r'$ for $r\geq 2$? Is it true that $c_r'<c_r$?
It is true that $c_1=c_1'$, and the classical bound on the size of Sidon sets (see [30]) implies $c_1=c_1'=1$.
For the analogous question with $n=a-b$ they prove that $\lvert A\rvert\sim N^{1/2}$.
This is a weaker form of [840].