Asks whether a graph with at most $k$ edge-disjoint triangles can be made triangle-free by deleting at most $2k$ edges (Tuza's conjecture). It is open, the erdosproblems.com label was falsifiable, and Haxell's $66/23$ is the best refereed constant.

• Graph theory

Asks for $\lim_{N\to\infty}F(N)/N$, where $F(N)$ is the size of the largest subset of $\{1,\ldots,N\}$ containing no set $\{n,2n,3n\}$, and whether this limit is irrational.

• Additive combinatorics

Asks for estimates of $f(k)$, the supremum of $\sum_{n\in A}1/n$ over sets $A$ of positive integers with no $k$-term arithmetic progression, and whether $f(k)/\log W(k)\to\infty$, where $W(k)$ is the van der Waerden number.

• Additive combinatorics
• Arithmetic progressions

Asks for the value of $\lim_{N\to\infty}F(N)/N^{1/2}$, where $F(N)$ is the smallest size of a set $A\subseteq\{0,1,\ldots,N\}$ with $\{0,1,\ldots,N\}\subseteq A-A$.

• Additive combinatorics

Asks for the smallest $h(d)$ such that one function $f:\mathbb{N}\to\{-1,1\}$ has $\lvert\sum_{n\in P}f(n)\rvert\le h(d)$ for every $d\ge1$ and every finite arithmetic progression $P$ with common difference $d$.

• Discrepancy
• Arithmetic progressions

Asks for the best $f(d)$ such that every $2$-coloring of the integers has, for infinitely many $d$, a monochromatic arithmetic progression of difference $d$ and length $f(d)$. Beck (1980) gives $f(d)\le(1+o(1))\log_2 d$, and no usable lower bound is known.

• Additive combinatorics
• Ramsey theory
• Arithmetic progressions

Asks for the least $k$ such that the plane can be colored red and blue with no two red points at unit distance and no blue $k$-term progression with step $1$. If the step is left free, Erdős and Graham's question has no finite answer.

• Geometry
• Ramsey theory

Asks for the largest $k$ such that every permutation of the integers contains a monotone $k$-term arithmetic progression.

• Arithmetic progressions

Asks whether the positive integers can be split into two sets, each of which can be permuted to avoid monotone three-term arithmetic progressions.

• Arithmetic progressions

Asks whether the longest arithmetic progression of primes in $\{1,\ldots,N\}$ has length $o(\log N)$.

• Primes
• Arithmetic progressions

Asks how large a subset free of $k$-term arithmetic progressions can be guaranteed in any set of $N$ integers, and how that size compares with the case of $\{1,\ldots,N\}$.

• Additive combinatorics
• Arithmetic progressions

Asks whether some integer $m\ge1$ coprime to $6$ makes $2^k3^\ell m+1$ composite for all $k,\ell\ge0$.

• Primes
• Covering systems

Asks whether the gap after the squarefree number $s_n$ is $\ll_\epsilon s_n^{\epsilon}$ for every $\epsilon>0$, and whether it is at most $(1+o(1))\frac{\pi^2}{6}\frac{\log s_n}{\log\log s_n}$.

• Number theory

Asks whether the plane contains a dense set of points all of whose pairwise distances are rational.

• Geometry
• Distances

Asks whether, for every $n\ge4$, there are $n$ points in the plane, no three on a line and no four on a circle, with all pairwise distances integers.

• Geometry
• Distances

Asks for which $n$ there are $n$ points in the plane, no three on a line and no four on a circle, that determine $n-1$ distinct distances, the $i$th of which occurs $i$ times.

• Geometry
• Distances

Asks whether the $n$ with $d_{n+1}\ge d_n$ and those with $d_{n+1}\le d_n$ each have density $1/2$, where $d_n=p_{n+1}-p_n$, and whether $d_{n+1}=d_n$ for infinitely many $n$.

• Number theory
• Primes

Asks for good upper and lower bounds on the gaps between consecutive integers that are sums of two squares.

• Number theory
• Squares

Asks whether $\sum_{n\le N}d_n^2\ll N(\log N)^2$, where $d_n=p_{n+1}-p_n$ is the gap after the $n$th prime.

• Number theory
• Primes

Asks whether, for every $c\ge0$, the density $f(c)$ of integers $n$ with $p_{n+1}-p_n<c\log n$ exists and is a continuous function of $c$.

• Number theory
• Primes

Asks whether the number $f(n)$ of solutions of $n=p+2^k$, with $p$ prime and $k\ge0$, satisfies $f(n)=o(\log n)$.

• Number theory
• Primes

Asks whether, for any $c_1,c_2>0$ and all large $x$, there are more than $c_1\log x$ consecutive primes $\le x$ whose pairwise differences all exceed $c_2$.

• Number theory
• Primes

Asks whether the largest subset of $\{1,\ldots,N\}$ whose triple sums $a+b+c$ are all distinct has size asymptotic to $N^{1/3}$.

• Additive combinatorics
• Sidon sets

Asks whether every increasing integer sequence with $a_n/a_{n-1}^2\to1$ and $\sum 1/a_n$ rational must satisfy $a_n=a_{n-1}^2-a_{n-1}+1$ for all large $n$.

• Number theory
• Irrationality

Asks whether, for $C>1$, the integers of the form $p+\lfloor C^k\rfloor$, with $p$ prime and $k\ge0$, have positive density.

• Number theory
• Primes