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3 solved out of 12 shown
Schoenberg proved that for every $c\in [0,1]$ the density of \[\{ n\in \mathbb{N} : \phi(n)<cn\}\] exists. Let this density be denoted by $f(c)$. Is it true that there are no $x$ such that $f'(x)$ exists and is positive?
Let $A$ be a finite set of integers. Is it true that for every $\epsilon>0$ \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg_\epsilon \lvert A\rvert^{2-\epsilon}?\]
The sum-product problem. Erdős and Szemerédi [ErSz83] proved a lower bound of $\lvert A\rvert^{1+c}$ for some constant $c>0$, and an upper bound of $o(\lvert A\rvert^2)$. The lower bound has been improved a number of times. The current record is \[\max( \lvert A+A\rvert,\lvert AA\rvert)\gg\lvert A\rvert^{\frac{1558}{1167}-o(1)}\] due to Rudnev and Stevens [RuSt22] (note $1558/1167=1.33504\cdots$).
A set of integers $A$ is Ramsey $r$-complete if, whenever $A$ is $r$-coloured, all sufficiently large integers can be written as a monochromatic sum of elements of $A$. Prove any non-trivial bounds about the growth rate of such an $A$ for $r>2$.
A paper of Burr and Erdős [BuEr85] proves both upper and lower bounds for $r=2$, showing that there exists some $c>0$ such that it cannot be true that \[\lvert A\cap \{1,\ldots,N\}\rvert \leq c(\log N)^2\] for all large $N$, and also constructing a Ramsey $2$-complete $A$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert \ll (\log N)^3.\] Burr has shown that the sequence of $k$th powers is Ramsey $r$-complete for every $r,k\geq 1$.

Solved by Conlon, Fox, and Pham [CFP21], who constructed for every $r\geq 2$ an $r$-Ramsey complete $A$ such that for all large $N$ \[\lvert A\cap \{1,\ldots,N\}\rvert \ll r(\log N)^2,\] and showed that this is best possible, in that there exists some constant $c>0$ such that if $A\subset \mathbb{N}$ satisfies \[\lvert A\cap \{1,\ldots,N\}\rvert \leq cr(\log N)^2\] for all large $N$ then $A$ cannot be $r$-Ramsey complete.

Additional thanks to: Mehtaab Sawhney
Find the value of $\lim_{k\to \infty}R(k)^{1/k}$.
Erdős offers \$100 for just a proof of the existence of this constant, without determining its value. He also offers \$1000 for a proof that the limit does not exist, but says 'this is really a joke as [it] certainly exists'. Erdős proved \[\sqrt{2}\leq \liminf_{k\to \infty}R(k)^{1/k}\leq \limsup_{k\to \infty}R(k)^{1/k}\leq 4.\] The upper bound has been improved to $4-\tfrac{1}{128}$ by Campos, Griffiths, Morris, and Sahasrabudhe [CaGrMoSa23].
Let $a,b,c$ be three integers which are pairwise coprime. Is every large integer the sum of distinct integers of the form $a^kb^lc^m$ ($k,l,m\geq 0$), none of which divide any other?
Conjectured by Erdős and Lewin [ErLe96], who (among other related results) prove this when $a=3$, $b=5$, and $c=7$.
Let $f(n)$ be maximal such that if $A\subseteq\mathbb{N}$ has $\lvert A\rvert=n$ then $\prod_{a\neq b\in A}(a+b)$ has at least $f(n)$ distinct prime factors. Is it true that $f(n)/\log n\to\infty$?
Investigated by Erdős and Turán [ErTu34] in their first joint paper, where they proved that \[\log n \ll f(n) \ll n/\log n\] (the upper bound is trivial, taking $A=\{1,\ldots,n\}$). Erdős says that $f(n)=o(n/\log n)$ has never been proved, but perhaps never seriously attacked.
Let $G$ be a graph with $10n$ vertices such that every subgraph on $5n$ vertices has more than $2n^2$ many edges. Must $G$ contain a triangle?
A blown-up $C_5$ (or, as Simonovits observed, a blown-up Petersen graph) shows that this would be best possible.
Additional thanks to: Boris Alexeev
Let $A\subset \mathbb{R}^2$ be a set of $n$ points such that any subset of size $4$ determines at least $5$ distinct distances. Must $A$ determine $\gg n^2$ many distances?
Erdős also makes the even stronger conjecture that $A$ must contain $\gg n$ many points such that all pairwise distances are distinct.
The density of integers which have two divisors $d_1,d_2$ such that $d_1<d_2<2d_1$ exists and is equal to $1$.
The answer is yes (in fact with $2$ replaced with any constant $c>1$), proved by Maier and Tenenbaum [MaTe84].

See also [449].

Give an asymptotic formula for $R(3,k)$.
It is known that there exists some constant $c>0$ such that for large $k$ \[c\frac{k^2}{\log k}\leq R(3,k) \leq (1+o(1))\frac{k^2}{\log k}.\] The lower bound is due to Kim [Ki95], the upper bound is due to Shearer [Sh83], improving an earlier bound of Ajtai, Komlós, and Szemerédi [AjKoSz80]. The lower bound has been improved to \[\left(\frac{1}{4}-o(1)\right)\frac{k^2}{\log k}\] independently by Bohman and Keevash [BoKe21] and Pontiveros, Griffiths and Morris [PGM20]. The latter collection of authors conjecture that this lower bound is the true order of magnitude.
Prove that \[R(4,k) \gg \frac{k^3}{(\log k)^{O(1)}}.\]
Spencer [Sp77] proved \[R(4,k) \gg (k\log k)^{5/2}.\] Ajtai, Komlós, and Szemerédi [AjKoSz80] proved \[R(4,k) \ll \frac{k^3}{(\log k)^2}.\] This is true, and was proved by Mattheus and Verstraete [MaVe23], who showed that \[R(4,k) \gg \frac{k^3}{(\log k)^4}.\]
Let $R(3;k)$ be the minimal $n$ such that if the edges of $K_n$ are coloured with $k$ colours then there must exist a monochromatic triangle. Determine \[\lim_{k\to \infty}R(3;k)^{1/k}.\]
Erdős offers \$100 for showing that this limit is finite. An easy pigeonhole argument shows that \[R(3;k)\leq 2+k(R(3;k-1)-1),\] from which $R(3;k)\leq \lceil e k!\rceil$ immediately follows. The best-known upper bounds are all of the form $ck!+O(1)$, and arise from this type of inductive relationship and computational bounds for $R(3;k)$ for small $k$. The best-known lower bound (coming from lower bounds for Schur numbers) is due to Exoo [Ex94], \[R(3;k) \gg (321)^{k/5}.\]

See also [483].

Additional thanks to: Antonio Girao, David Penman