Asks whether bounded clique number and large chromatic number force two anticomplete vertex sets, both of large chromatic number. This is the El-Zahar–Erdős problem, open beyond the case of chromatic number $3$.

• Graph theory

Asks whether some Sierpinski number, an odd $m$ with $2^km+1$ never prime, has no finite set of primes containing a divisor of every $2^km+1$.

• Number theory
• Covering systems

Asks for the shortest length of a path from $0$ to $|z|=1$ inside $\{z:|f(z)|\le1\}$, where $f$ is a monic polynomial of degree $n$ with all roots in $|z|\le1$.

• Analysis

Asks for the minimum over $x_1,\ldots,x_n\in[-1,1]$ of $I=\int_{-1}^1\sum_k\lvert l_k(x)\rvert^2\,dx$, where the $l_k$ are the Lagrange basis polynomials, and whether $\min I=2-(1+o(1))/n$.

• Analysis
• Polynomials

For a sequence of nodes in $[-1,1]$, let $L_n(x)=\sum_{k\le n}\lvert l_k(x)\rvert$. Asks whether some $x\in(-1,1)$ has $L_n(x)>\frac{2}{\pi}\log n-O(1)$ for infinitely many $n$, and whether $\limsup_n L_n(x)/\log n\ge 2/\pi$ for almost all $x$.

• Analysis
• Polynomials

Asks whether every orbit of the shortcut Collatz map reaches $1$; open, with verification below $2^{71}$, no nontrivial cycle with at most $91$ local minima, and Tao's almost-all theorem. erdosproblems.com adds a caveat on the reported Erdős prize figure.

• Number theory
• Iterated functions

Asks whether, with $d_n=p_{n+1}-p_n$, the ratio $\max_{n<x}d_nd_{n-1}/(\max_{n<x}d_n)^2$ tends to $0$ as $x\to\infty$.

• Number theory
• Primes

Asks whether infinitely many $n$, or any $n>105$, have $n-2^k$ prime for every $k$ with $1<2^k<n$.

• Number theory
• Primes

Asks to estimate, for given primes $p_1<\cdots<p_u$, how many integers divisible by some $p_i$ every interval of $k$ consecutive positive integers must contain.

• Number theory
• Primes

Asks whether $\limsup 1_A\ast 1_B(n)=\infty$ whenever $A=\{a_1<a_2<\cdots\}$ and $B=\{b_1<b_2<\cdots\}$ are sets of positive integers with $a_n/b_n\to1$ and $A+B$ contains all large integers.

• Additive combinatorics
• Additive bases

Asks whether $\{2^m3^n : m,n\ge0\}$ is an essential component, that is, a set $A$ with $d_s(A+B)>d_s(B)$ whenever the Schnirelmann density $d_s(B)$ lies strictly between $0$ and $1$.

• Number theory

Asks whether, for every $\alpha\in[0,1]$, some ring or field in $\mathbb{R}$ has Hausdorff dimension $\alpha$. Mauldin's theorem gives yes under the continuum hypothesis, so ZFC cannot refute the existence.

• Analysis

Asks whether the chromatic number of a random graph on $n$ vertices, each edge present independently with probability $1/2$, is almost surely concentrated on boundedly many values, and whether it is within a slowly growing $\omega(n)$ of any given $f(n)$ with probability below $1/2$.

• Graph theory
• Chromatic number

Asks for the most edges an $r$-graph on $n$ vertices can have with no $k$ vertices spanning $s$ edges, the Brown–Erdős–Sós problem. The associated conjecture is proved for $s=3$ and, in its linear form, for large uniformity, but open for $3$-graphs with $k=s+3$ and $s\ge4$. A 2026 preprint claims the asymptotics whenever $(rs-k)/(s-1)$ is an integer $t$ with $2\le t<r$, except for $3$-graphs with even $s$.

• Hypergraphs
• Turán numbers

Asks whether Erdős's 1964 upper exponent for the Turán number of $K_t(r)$, the complete $t$-partite $t$-uniform hypergraph with $r$ vertices per class, is attained up to $o(1)$. It is known for $t=2$ and $r\le3$, and a 2026 preprint claims the case $t=3$, $r=2$.

• Hypergraphs
• Turán numbers

Asks whether there is a constant $C>1$ such that every finite projective plane has a set of points meeting each line in at least $1$ and at most $C$ points.

• Combinatorics

Asks whether the number $g(n)$ of groups of order $n$ satisfies $g(n)\leq g(2^m)$ whenever $n\leq 2^m$.

• Group theory

Asks for an asymptotic formula for the number of subgroups of the symmetric group $S_n$, and for statistical results on their orders.

• Group theory

Asks for a statistical description of the arithmetic structure of the orders of subgroups of the symmetric group $S_n$.

• Group theory

Asks whether, for infinite $\lambda$, $2^\lambda\to(\kappa_\alpha+1)_{\alpha<\gamma}^{r+1}$ implies $\lambda\to(\kappa_\alpha)_{\alpha<\gamma}^{r}$; open under the Erdős–Hajnal list's conditions $\gamma\geq2$ and all $\kappa_\alpha>r$, which exclude the failures of the erdosproblems.com wording.

• Set theory
• Ramsey theory

Asks for a proof of $\aleph_{\omega+1}\not\to(\aleph_{\omega+1},3,\ldots,3)_{\aleph_0}^2$ without the generalized continuum hypothesis.

• Set theory
• Ramsey theory

Asks whether the negative partition relation $\omega_1^2\not\to(\omega_1^2,3)^2$ holds.

• Set theory
• Ramsey theory

Asks whether it is consistent that $\omega_2\to(\alpha)_2^2$ for every $\alpha<\omega_2$.

• Set theory
• Ramsey theory

Asks whether $\omega_1^2\to(\omega_1\omega,3,\ldots,3)_{k+1}^2$ holds for every finite $k$.

• Set theory
• Ramsey theory

Asks whether several partition relations for pairs on the ordinals $\omega_2$ and $\omega_3$ hold under the generalized continuum hypothesis, or are consistent with it.

• Set theory
• Ramsey theory