Asks for the largest size of a set $A\subseteq\mathbb{R}^d$ in which every three points form an isosceles triangle.

• Geometry
• Distances

Asks to estimate the smallest area $\alpha(n)$ such that every set of $n$ points in the unit disk contains three points forming a triangle of area at most $\alpha(n)$.

• Geometry

Asks for the fewest colors needed to color the plane so that no two points at distance $1$ share a color.

• Geometry
• Ramsey theory

Asks whether, for every monic nonconstant $f\in\mathbb{C}[z]$, the set $\{z:\lvert f(z)\rvert\le 1\}$ can be covered by circles whose radii sum to at most $2$.

• Analysis
• Polynomials

Asks whether every set $A$ of $N$ positive integers has some $\theta$ with $\sum_{n\in A}\cos(n\theta)<-cN^{1/2}$, for an absolute constant $c>0$. The erdosproblems.com wording, over all integers, fails for sets containing $0$.

• Analysis

Asks for the greatest possible value of $\liminf_{r\to\infty}\max_n\lvert a_nr^n\rvert/\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ over transcendental entire functions $f=\sum a_nz^n$.

• Analysis

Asks whether an entire power series $\sum a_kz^{n_k}$ with $n_k/k\to\infty$ must take every complex value infinitely often.

• Analysis

Asks for the limiting growth rate per step, $C_k=\lim_{n\to\infty}f(n,k)^{1/n}$, of the number $f(n,k)$ of self-avoiding walks of $n$ steps from the origin in $\mathbb{Z}^k$.

• Geometry

Asks whether the expected end-to-end distance $d_k(n)$ of an $n$-step self-avoiding walk in $\mathbb{Z}^k$ satisfies $d_2(n)/n^{1/2}\to\infty$, and $d_k(n)\ll n^{1/2}$ for every $k\ge3$.

• Geometry
• Probability

Asks for the order of $\ell(N)$, the size of the largest Sidon subset, one with no non-trivial equal pairwise sums, guaranteed inside every set of $N$ reals.

• Number theory
• Sidon sets

Asks for an estimate of the least $N=F(k)$ such that every two-coloring of $\{1,\ldots,N\}$ has a $k$-element set whose non-empty subset sums all have the same color.

• Number theory
• Ramsey theory

Asks for estimates of $f_r(N)$, the size of the largest subset of $\{1,\ldots,N\}$ with no $r$ elements whose pairwise greatest common divisors are all equal, for $r\ge3$.

• Number theory

Asks for estimates of $h(n)$, the least size of $\{a/\gcd(a,b):a,b\in A\}$ over all sets $A\subseteq\mathbb{N}$ of size $n$.

• Additive combinatorics
• Number theory

Asks whether the gap $R(3,k+1)-R(3,k)$ between consecutive Ramsey numbers of a triangle against a complete graph tends to infinity, and whether it is $o(k)$. Both are open, and the gap is known only to lie between $3$ and $k+1$.

• Graph theory
• Ramsey theory

Asks whether, for all large $m$, the graph with $m$ edges that is as complete as possible has the largest Ramsey number among graphs with $m$ edges and no isolated vertices; open. The erdosproblems.com wording, for every $m$, fails at two edges.

• Graph theory
• Ramsey theory

Asks for the Ramsey number $R(C_4,S_n)$ of a four-cycle against the star with $n$ edges, and whether, for every $c>0$, infinitely many $n$ have $R(C_4,S_n)\le n+\sqrt{n}-c$.

• Graph theory
• Ramsey theory

Asks for a proof that $R_k(C_{2n+1})/R_k(K_3)\to0$ as $k\to\infty$ for every $n\ge2$, where $R_k$ is the $k$-color Ramsey number.

• Graph theory
• Ramsey theory

Asks for the $k$-color Ramsey number $R_k(C_{2n})$ of the even cycle on $2n$ vertices.

• Graph theory
• Ramsey theory

Asks whether, for every $n>3$, every 3-coloring of the edges of $K_{4n-3}$ has a monochromatic cycle of length $n$; the bound is known for all large $n$. The erdosproblems.com wording also includes the triangle, where it fails.

• Graph theory
• Ramsey theory

Asks for the $k$-color Ramsey number $R_k(K_{s,t})$ of the complete bipartite graph with $s$ vertices in one class and $t$ in the other.

• Graph theory
• Ramsey theory

Asks for the size Ramsey number of $K_{n,n}$, the complete bipartite graph with $n$ vertices on each side; its order is known to lie between $n^22^n$ and $n^32^n$.

• Graph theory
• Ramsey theory

Asks for a proof of the 1978 formula $\hat{R}(F_1,F_2)=\sum_{2\le k\le s+t}l_k$, with $l_k=\max\{n_i+m_j-1:i+j=k\}$, for the star forests $F_1=\bigcup_{i\le s}K_{1,n_i}$ and $F_2=\bigcup_{j\le t}K_{1,m_j}$ with stars in decreasing order of size; proved in special cases only.

• Graph theory
• Ramsey theory

Asks for a proof that $\log_{r-1}R_r(n)\asymp_r n$ for $r\ge3$, where $R_r(n)$ is the two-color Ramsey number of the complete $r$-uniform hypergraph on $n$ vertices and $\log_{r-1}$ is the $(r-1)$-fold iterated logarithm.

• Graph theory
• Ramsey theory
• Hypergraphs

Asks for a proof that $F(n,\alpha)\sim c_\alpha\log n$ for each $0\le\alpha<1/2$, where $F(n,\alpha)$ is the least $m$ such that some two-coloring of $K_n$ gives every set $X$ of at least $m$ vertices more than $\alpha\binom{|X|}{2}$ edges of each color.

• Graph theory
• Ramsey theory
• Hypergraphs

Asks whether $R_3(n)\ge2^{2^{cn}}$ for some $c>0$, where $R_3(n)$ is the two-color Ramsey number of the complete $3$-uniform hypergraph on $n$ vertices.

• Graph theory
• Ramsey theory
• Hypergraphs