Open-problem statements, with their sources attached.

The smallest integer which is unknown to be in $A - A$ is $33$.

It may be true that all or almost all integers are in $A - A$.

It may be true that all or almost all integers are in $A - A$.

Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence?

Does Ulam's sequence eventually have periodic differences? That is, is $a(n+1) - a(n)$ eventually periodic?

Part (iii), is the density of the sequence 0?

When $A =\{a_1 < \cdots\}$ corresponds to the set of primes, it is conjectured that the $\limsup$ of the number of representations $n=\sum_{u\leq i\leq v}a_i$ is infinite.

When $A =\{a_1 < \cdots\}$ corresponds to the set of primes, it is conjectured that the set of numbers $n$ that have representations $n=\sum_{u\leq i\leq v}a_i$ has positive upper density.

It is conjectured that if $A =\{a_1 < \cdots\}$ and $g$ counts the number of representations $n=\sum_{u\leq i\leq v}a_i$ such that the sum has at least two terms, then for all $n$ we have $1 \leq g(n)$ for sufficiently large $n$.

Find a better lower bound!

Find a better upper bound!

Find the value of the limit of MinOverlapQuotient!

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).

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.

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

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 $m>n\geq \max(A)$, $\frac{\lvert B\cap [1,m]\rvert }{m}< 2\frac{\lvert B\cap [1,n]\rvert}{n}?$

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 $r \geq 3$, 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].

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 $N$.

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 $f_r(N)$.