Open-problem statements, with their sources attached.

A variant asking for explicit bounds on how large N needs to be in terms of M.

This version provides a constructive function f such that for all M ≥ 1 and N ≥ f(M), every maximal Sidon set A ⊆ {1,…,N} has another Sidon set B ⊆ {1,…,N} of size M with disjoint difference sets (apart from 0).

Case 1: Does $A + B$ have zero upper and lower density?

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}?

This problem asks whether any Sidon set can be extended to achieve a density arbitrarily close to the optimal density for Sidon sets.

Case 2: Does $A + B$ have zero lower density, but positive upper density?

The case where we start with an empty set (constructing large Sidon sets).

Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $|A| = n$ then $\prod_{a\neq b\in A}(a + b)$ has at least $f(n)$ distinct prime factors. Is it true that $\frac{f(n)}{\log n} \to\infty$?

Let $A$ be a finite set and $B=\{ n \geq 1 : a\mid n\textrm{ for some }a\in A\}.$ Is it true that, for every $m>n\geq \max(A)$, $\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?$

Erdős says that $f(n) = o(\frac{n}{\log n})$ has never been proved.

For every $x \in \mathbb{R}$ let $A_x \subset \mathbb{R}$ be a bounded set with outer measure $< 1$. Must there exist an infinite independent set, that is, some infinite $X \subseteq \mathbb{R}$ such that $x \notin A_y$ for all $x \neq y \in X$?

If the sets $A_x$ are closed and have measure $< 1$, then must there exist an independent set of size $3$?

Known results: Erdős–Hajnal [ErHa60] proved the existence of arbitrarily large finite independent sets. Hechler [He72] showed the answer is no assuming the continuum hypothesis.

Let $k\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is, must there always exist a prime $p\mid N$ such that $p^2\nmid N$?

Let $r \geq 3$, and let $f_r(N)$ denote the size of the largest subset of $\{1,\ldots,N\}$ such that no subset of size $r$ has the same pairwise greatest common divisor between all elements. Erdős [Er64] proved that $f_3(N) > N^{c/\log\log N}$ for some constant $c > 0$, and conjectured this should also be an upper bound; here we state the conjectural upper bound for all $r \geq 3$.

See also [536].

Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all positive integers $m$, there must be at least $k$ distinct primes $p$ such that $p\mid m(m+1)\cdots (m+n)$ and yet $p^2$ does not divide the right hand side.

[Er82c] Erdős, Paul, "Miscellaneous problems in number theory". Congr. Numer. (1982), 25-45.,

The first open case of Erdős Problem 535 is $r = 3$: there should exist $c > 0$ such that $f_3(N) \leq N^{c/\log\log N}$ for all sufficiently large $N$.

In [Er80] Erdős asks whether $\lim_{k \to \infty} (W(k))^{1/k} = \infty$

Erdős [Er73] records that Abbott pointed out the ordinary sunflower conjecture does not seem to suffice here. The stronger auxiliary conjecture uses $Ω(n)=k$, i.e. prime factors counted with multiplicity; this stronger statement would imply the conjectured upper bound for $f_r(N)$.

In [Er81] Erdős asks whether $\frac{W(k+1)}{W(k)} \to \infty$.

Let $h(n)$ be maximal such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set$\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$has size at least $h(n)$. Estimate $h(n)$.

In [Er80] Erdős asks whether $W(k)/2^k\to \infty$.

Let $h(n)$ be maximal such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set$\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$has size at least $h(n)$. Is $h(n) = \Theta(\sqrt{n})$?

Let $A ⊆ \mathbb{N}$. Let $B ⊆ \mathbb{N}$ 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 $\epsilon > 0$ and large $N$, $|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}$?