174 problems

I conjectured long ago that if … then the $a_i$'s contain arbitrarily long arithmetic progressions. If true this of course implies that there are arbitrarily long arithmetic progre…

Must any ordering of the reals contain a monotone $k$-term arithmetic progression for every $k$?

The assertion is false: it is not true that every Sidon set $A\subseteq\mathbb{N}$ has an infinite arithmetic progression $Y\subseteq\mathbb{N}$ with…

Let $N(k,\ell)$ be the minimal $N$ such that for any $f:\{1,\ldots,N\}\to\{-1,1\}$ there must exist a $k$-term arithmetic progression $P$ such that…

Let $A(n;k)$ be the minimum, over all two-colourings of $\{1,\ldots,n\}$, of the number of monochromatic $k$-term arithmetic progressions. Determine an asymptotic formula for…

The assertion is false: it is not true that every set $A\subseteq\mathbb{R}$ containing no three-term arithmetic progression has an infinite arithmetic progression…

Let $(a_i)_{i\geq1}$ be an infinite sequence in $\mathbb Z^d$ such that $a_{i+1}-a_i$ is always a positive coordinate unit vector. For which dimensions $d$ must the sequence contai…

For each $k≥3$, does every finite colouring of the positive integers contain a monochromatic $k$-term arithmetic progression all of whose terms are prime? Does it contain a monochr…

For every $δ>0$ and integer $k≥3$, is there an $m=m(δ,k)$ such that, for all sufficiently large $N$, whenever $A,B⊆{1,…,N}$ satisfy $|A|≥δN$ and $|B|≥m$, the set $A$ contains a non…

Can the positive integers be coloured with two colours so that, for every $ε>0$, every monochromatic arithmetic progression with first term $a$ has length $O_ε(a^ε)$?

Let $N\geq 1$. What is the largest $t$ such that there are $A_1,\ldots,A_t\subseteq \{1,\ldots,N\}$ with $A_i\cap A_j$ a non-empty arithmetic progression for all $i\neq j$?

Let $A(n)=\{a_0<a_1<\cdots\}$ be the sequence defined by $a_0=0$ and $a_1=n$, and for $k\geq 1$ define $a_{k+1}$ as the least positive integer such that there is no three-term arit…

A set $s\subseteq\mathbb{N}$ is a prime arithmetic progression if every $p\in s$ is prime and there exists a natural number $l>0$ such that $s$ is an arithmetic progression of leng…

Let $G_k(N)$ be such that any set of $N$ integers contains a subset of size at least $G_k(N)$ which does not contain a $k$-term arithmetic progression. Determine the size of…

Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?

We conclude this topic with a very annoying question: Is it possible to partition $\mathbf{Z}^+$ into two sets, each of which can be permuted to avoid monotone 3-term A.P.'s? If we…

The question of whether or not monotone 4-term A.P.'s must occur is currently completely open.

What is the largest natural number $k$ such that every permutation of $\mathbb Z$ contains a monotone $k$-term arithmetic progression? Equivalently, determine … Here “contains in m…

Let $H(k)$ be the smallest $N$ such that in any finite colouring of $\{1,\ldots,N\}$ (into any number of colours) there is always either a monochromatic $k$-term arithmetic progres…

For integers $r<s$, let $f_r(N;s)$ be the least number such that every sequence of $N$ integers containing at least $f_r(N;s)$ arithmetic progressions of length $r$ also contains o…

Find the smallest $h(d)$ such that the following holds. There exists a function $f:\mathbb{N}\to\{-1,1\}$ such that, for every $d\geq 1$,…

Let $k\geq 3$ and $f(k)$ be the supremum of $\sum_{n\in A}\frac{1}{n}$ as $A$ ranges over all sets of positive integers which do not contain a $k$-term arithmetic progression. Esti…

Let $h(N)$ be the smallest $k$ such that $\{1,\ldots,N\}$ can be coloured with $k$ colours so that every four-term arithmetic progression must contain at least three distinct colou…

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

Let $k\geq 3$. Are there $k$ consecutive primes in arithmetic progression?