OPEN - $500

If $A\subseteq \{1,\ldots,N\}$ with $\lvert A\rvert=n$ is such that the subset sums $\sum_{a\in S}a$ are distinct for all $S\subseteq A$ then
\[N \gg 2^{n}.\]

Erdős called this 'perhaps my first serious problem'. The powers of $2$ show that $2^n$ would be best possible here. The trivial lower bound is $N \gg 2^{n}/n$, since all $2^n$ distinct subset sums must lie in $[0,Nn)$. Erdős and Moser [Er56] proved
\[ N\geq (\tfrac{1}{4}-o(1))\frac{2^n}{\sqrt{n}}.\]
(In [Er85c] Erdős offered \$100 for any improvement of the constant $1/4$ here.)

A number of improvements of the constant have been given (see [St23] for a history), with the current record $\sqrt{2/\pi}$ first proved in unpublished work of Elkies and Gleason. Two proofs achieving this constant are provided by Dubroff, Fox, and Xu [DFX21], who in fact prove the exact bound $N\geq \binom{n}{\lfloor n/2\rfloor}$.

In [Er73] and [ErGr80] the generalisation where $A\subseteq (0,N]$ is a set of real numbers such that the subset sums all differ by at least $1$ is proposed, with the same conjectured bound. (The second proof of [DFX21] applies also to this generalisation.)

This problem appears in Erdős' book with Spencer [ErSp74] in the final chapter titled 'The kitchen sink'. As Ruzsa writes in [Ru99] "it is a rich kitchen where such things go to the sink".

The sequence of minimal $N$ for a given $n$ is A276661 in the OEIS.

See also [350].

OPEN - $1000

Let $f(n,k)$ be minimal such that every $\mathcal{F}$ family of $n$-uniform sets with $\lvert F\rvert \geq f(n,k)$ contains a $k$-sunflower. Is it true that
\[f(n,k) < c_k^n\]
for some constant $c_k>0$?

Erdős and Rado [ErRa60] originally proved $f(n,k)\leq (k-1)^nn!$. Kostochka [Ko97] improved this slightly (in particular establishing an upper bound of $o(n!)$, for which Erdős awarded him the consolation prize of \$100), but the bound stood at $n^{(1+o(1))n}$ for a long time until Alweiss, Lovett, Wu, and Zhang [ALWZ20] proved
\[f(n,k) < (Ck\log n\log\log n)^n\]
for some constant $C>1$. This was refined slightly, independently by Rao [Ra20], Frankston, Kahn, Narayanan, and Park [FKNP19], and Bell, Chueluecha, and Warnke [BCW21], leading to the current record of
\[f(n,k) < (Ck\log n)^n\]
for some constant $C>1$.

In [Er81] offered \$1000 for a proof or disproof even just in the special case when $k=3$, which he expected 'contains the whole difficulty'. He also wrote 'I really do not see why this question is so difficult'.

The usual focus is on the regime where $k=O(1)$ is fixed (say $k=3$) and $n$ is large, although for the opposite regime Kostochka, Rödl, and Talysheva [KoRoTa99] have shown \[f(n,k)=(1+O_n(k^{-1/2^n}))k^n.\]

OPEN - $500

If $A\subseteq \mathbb{N}$ is such that $A+A$ contains all but finitely many integers then $\limsup 1_A\ast 1_A(n)=\infty$.

Conjectured by Erdős and Turán. They also suggest the stronger conjecture that $\limsup 1_A\ast 1_A(n)/\log n>0$.

Another stronger conjecture would be that the hypothesis $\lvert A\cap [1,N]\rvert \gg N^{1/2}$ for all large $N$ suffices.

Erdős and Sárközy conjectured the stronger version that if $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ with $a_n/b_n\to 1$ are such that $A+B=\mathbb{N}$ then $\limsup 1_A\ast 1_B(n)=\infty$.

See also [40].

OPEN - $1000

Let $h(N)$ be the maximum size of a Sidon set in $\{1,\ldots,N\}$. Is it true that, for every $\epsilon>0$,
\[h(N) = N^{1/2}+O_\epsilon(N^\epsilon)?\]

A problem of Erdős and Turán. It may even be true that $h(N)=N^{1/2}+O(1)$, but Erdős remarks this is perhaps too optimistic. Erdős and Turán [ErTu41] proved an upper bound of $N^{1/2}+O(N^{1/4})$, with an alternative proof by Lindström [Li69]. Both proofs in fact give
\[h(N) \leq N^{1/2}+N^{1/4}+1.\]
Balogh, Füredi, and Roy [BFR21] improved the bound in the error term to $0.998N^{1/4}$, which has been further optimised by O'Bryant [OB22] to yield
\[h(N)\leq N^{1/2}+0.99703N^{1/4}\]
for sufficiently large $N$.

OPEN - $100

Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial coincidences). Is it true that
\[ f(N)\sim N^{1/3}?\]

Originally asked to Erdős by Bose. Bose and Chowla [BoCh62] provided a construction proving one half of this, namely
\[(1+o(1))N^{1/3}\leq f(N).\]
The best upper bound known to date is due to Green [Gr01],
\[f(N) \leq ((7/2)^{1/3}+o(1))N^{1/3}\]
(note that $(7/2)^{1/3}\approx 1.519\cdots$).

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]).

OPEN

Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that the products $ab$ are distinct for all $a<b$. Is there a constant $c$ such that
\[F(n)=\pi(n)+(c+o(1))n^{3/4}(\log n)^{-3/2}?\]

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 \[\lvert A\rvert \leq \pi(n)+O(n^{\frac{r+1}{2r}})?\]

Erdős [Er68] proved that there exist some constants $0<c_1\leq c_2$ such that
\[\pi(n)+c_1 n^{3/4}(\log n)^{-3/2}\leq F(n)\leq \pi(n)+c_2 n^{3/4}(\log n)^{-3/2}.\]
Surprisingly, if we consider the corresponding problem in the reals (so consider the largest $A\subset [1,x]$ such that for any distinct $a,b,c,d\in A$ we have $\lvert ab-cd\rvert \geq 1$) then Alexander proved that $\lvert A\rvert> x/8e$ is possible (disproving an earlier conjecture of Erdős [Er73] that $m=o(x)$). Alexander's construction seems to be unpublished, and I have no idea what it is.

SOLVED

Let $A,B\subseteq \{1,\ldots,N\}$ be such that all the products $ab$ with $a\in A$ and $b\in B$ are distinct. Is it true that
\[\lvert A\rvert \lvert B\rvert \ll \frac{N^2}{\log N}?\]

This would be best possible, for example letting $A=[1,N/2]\cap \mathbb{N}$ and $B=\{ N/2<p\leq N: p\textrm{ prime}\}$.

See also [425].

This is true, and was proved by Szemerédi [Sz76].

OPEN

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. Estimate $f_r(N)$.

Erdős [Er64] proved that
\[f_r(N) \leq N^{\frac{3}{4}+o(1)},\]
and Abbott and Hanson [AbHa70] improved this exponent to $1/2$. Erdős [Er64] proved the lower bound
\[f_3(N) > N^{\frac{c}{\log\log N}}\]
for some constant $c>0$, and conjectured this should also be an upper bound.

Erdős writes this is 'intimately connected' with the sunflower problem [20]. Indeed, the conjectured upper bound would follow from the following stronger version of the sunflower problem: estimate the size of the largest set of integers $A$ such that $\omega(n)=k$ for all $n\in A$ and there does not exist $a_1,\ldots,a_r\in A$ and an integer $d$ such that $(a_i,a_j)=d$ for all $i\neq j$ and $(a_i/d,d)=1$ for all $i$. The conjectured upper bound for $f_r(N)$ would follow if the size of such an $A$ must be at most $c_r^k$. The original sunflower proof of Erdős and Rado gives the upper bound $c_r^kk!$.

See also [536].

Erdős and Klein [Er38] proved $\mathrm{ex}(n;C_4)\asymp n^{3/2}$. Reiman [Re] proved
\[\frac{1}{2\sqrt{2}}\leq \lim \frac{\mathrm{ex}(n;C_4)}{n^{3/2}}\leq \frac{1}{2}.\]
Erdős and Rényi, and independently Brown, gave a construction that proved if $n=q^2+q+1$, where $q$ is a prime power, then
\[\mathrm{ex}(n;C_4)\geq\frac{1}{2}q(q+1)^2.\]
Coupled with the upper bound of Reiman this implies that $\mathrm{ex}(n;C_4)\sim\frac{1}{2}n^{3/2}$ for all large $n$. Füredi [Fu83] proved that if $q>13$ then
\[\mathrm{ex}(n;C_4)=\frac{1}{2}q(q+1)^2.\]

See also [572].

OPEN

Let $\epsilon>0$. Is there some set $A\subset \mathbb{N}$ of density $>1-\epsilon$ such that $a_1\cdots a_r=b_1\cdots b_s$ with $a_i,b_j\in A$ can only hold when $r=s$?

Similarly, can one always find a set $A\subset\{1,\ldots,N\}$ with this property of size $\geq (1-o(1))N$?

An example of such a set with density $1/4$ is given by the integers $\equiv 2\pmod{4}$.

Selfridge constructed such a set with density $1/e-\epsilon$ for any $\epsilon>0$: let $p_1<\cdots<p_k$ be a sequence of large consecutive primes such that \[\sum_{i=1}^k\frac{1}{p_i}<1<\sum_{i=1}^{k+1}\frac{1}{p_i},\] and let $A$ be those integers divisible by exactly one of $p_1,\ldots,p_k$.

For the second question the set of integers with a prime factor $>N^{1/2}$ give an example of a set with size $\geq (\log 2)N$. Erdős could improve this constant slightly.

OPEN

Let $F(n)$ be the maximum possible size of a subset $A\subseteq\{1,\ldots,N\}$ such that $a\nmid bc$ whenever $a,b,c\in A$ with $a\neq b$ and $a\neq c$. Is there a constant $c$ such that
\[F(n)=\pi(n)+(c+o(1))n^{2/3}(\log n)^{-2}?\]

Erdős [Er38] proved there exist constants $0<c_1\leq c_2$ such that
\[\pi(n)+c_1n^{2/3}(\log n)^{-2}\leq F(n) \leq \pi(n)+c_2n^{2/3}(\log n)^{-2}.\]

Erdős [Er69] gave a simple proof that $F(n) \leq \pi(n)+n^{2/3}$: we define a graph with vertex set the union of those integers in $[1,n^{2/3}]$ with all primes $p\in (n^{2/3},n]$. We have an edge $u\sim v$ if and only if $uv\in A$. It is easy to see that every $m\leq n$ can be written as $uv$ where $u\leq n^{2/3}$ and $v$ is either prime or $\leq n^{2/3}$, and hence there are $\geq \lvert A\rvert$ many edges. This graph contains no path of length $3$ and hence must be a tree and have fewer edges than vertices, and we are done. This can be improved to give the upper bound mentioned by using a subset of integers in $[1,n^{2/3}]$.

More generally, one can ask for such an asymptotic for the size of sets such that no $a\in A$ divides the product of $r$ distinct other elements of $A$, with the exponent $2/3$ replaced by $\frac{2}{r+1}$.

See also [425].

OPEN

Let $g(n)$ be the maximal size of $A\subseteq \{1,\ldots,n\}$ such that $\prod_{n\in S}n$ are distinct for all $S\subseteq A$. Is it true that
\[g(n) \leq \pi(n)+\pi(n^{1/2})+o\left(\frac{x^{1/2}}{\log n}\right)?\]

Erdős proved [Er66]
\[g(n) \leq \pi(n)+O\left(\frac{x^{1/2}}{\log n}\right).\]
This upper bound would be essentially best possible, since one could take $A$ to be all primes and squares of primes.

OPEN

Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $<k$ solutions to $m=a_1a_2$ with $a_1<a_2\in A$.

Estimate $g_k(n)$. In particular, is it true that \[g_k(n)=\frac{\log\log n}{\log n}n+(c+o(1))\frac{n}{(\log n)^2}\] for some constant $c$?

Erdős [Er64d] proved that if $2^{r-1}<k\leq 2^r$ then
\[g_k(n) \sim \frac{(\log\log n)^{r-1}}{(r-1)!\log n}n\]
(which is the asymptotic count of those integers $\leq n$ with $r$ distinct prime factors).

In particular the asymptotics of $g_k(n)$ are known; in this problem Erdős was asking about the second order terms. For $k=3$ he could prove the existence of some $0<c_1\leq c_2$ such that \[\frac{\log\log n}{\log n}n+c_1\frac{n}{(\log n)^2}\leq g_k(n)\leq \frac{\log\log n}{\log n}n+c_2\frac{n}{(\log n)^2}.\]

The special case $k=2$ is the subject of [425].