Open-problem statements, with their sources attached.

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 $N$, $|\{1,\ldots,N\} \setminus B| \gg_\epsilon N^{1/2 - \epsilon}$?

To prove erdos_539.variants.sq it suffices to show $h(n)\ll n^{1/2}$.

Is it possible that $|\{1,\ldots,N\} \setminus B| = o(N^\frac{1}{2})$?

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 $h(n) = \Theta(n^{2/3})$?

Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

To prove erdos_539.variants.sq_cube_root it suffices to show $n^{2/3}\ll h(n)$.

Show that $r_k(N) = o_k(N / \log N)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

From [Er73]: The determination of

will perhaps be not too difficult.

Find functions $f_k$, such that $r_k(N) = O_k(f_k)$, where $r_k(N)$ the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression.

Let $R_r(n)$ denote the $r$-uniform hypergraph Ramsey number: the minimal $m$ such that if we $2$-colour all edges of the complete $r$-uniform hypergraph on $m$ vertices then there must be some monochromatic copy of the complete $r$-uniform hypergraph on $n$ vertices.

Prove that, for $r \ge 3$, $\log_{r-1} R_r(n) \asymp_r n,$ where $\log_{r-1}$ denotes the $(r-1)$-fold iterated logarithm.

Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $3$-term arithmetic progression.

Let $R_3(n)$ be the minimal $m$ such that if the edges of the $3$-uniform hypergraph on $m$ vertices are $2$-coloured then there is a monochromatic copy of the complete $3$-uniform hypergraph on $n$ vertices.

Is there some constant $c>0$ such that $R_3(n) \geq 2^{2^{cn}}?$

Does this imply that

Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges. Is it true that, if $H$ has $m$ edges and no isolated vertices, then $\hat{r}(G,H) \ll m$?

In other words: if $G$ is sparse (every induced subgraph on $k$ vertices has $≤ 2k-3$ edges), is $G$ Ramsey size linear?

Or