Browse Problems

Suppose that $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\gt c \alpha^2$?

From [Gr24] "It is quite easy to show using Cauchy-Schwarz that there must be such a rectangle with area $\gg \alpha^2 (\log 1/\alpha)^{-1}$."

Let `A ⊂ R` be a set of positive measure. Does $A$ contain an affine copy of `{1, 1/2, 1/4, . . . }`?

Problem 9 (i): is $r_3(N) \ll N(\log N)^{-10}$? Solved in [BlSi20].

Problem 9 (ii): is $r_5(N) \ll N(\log N)^{-c}$?

Problem 9 (iii): is $r_4(\mathbf{F}_5^n) \ll N^{1-c}$, where $N=5^n$?

**Grimm's Conjecture** If $n, n+1, \dots, n+k-1$ are all composite numbers, then there are $k$ distinct primes $p_i$ such that $p_i$ divides $n + i$ for all $0 \le i \le k-1$.

**Grimm's Conjecture, weaker version** If $n, n+1, \dots, n+k-1$ are all composite numbers, then their product has at least $k$ distinct prime divisors.

**Gromov's Polynomial Growth Theorem** : A finitely generated group has polynomial growth if and only if it is virtually nilpotent.

There exists a Hadamard matrix for all $n = 4k$.

The smallest order for which no Hadamard matrix is presently known is $668 = 4 * 167$.

For all $k ≤ 166$, it is known there that there is a Hadamard matrix of size $4 * k$.

The "chromatic number of the plane" is at least 4. This can be proven by considering the [Moser-Spindel graph](https://de.wikipedia.org/wiki/Moser-Spindel) or the [Golomb graph](https://en.wikipedia.org/wiki/Golomb_graph) graph.

Aubrey de Grey improved the lower bound for the chromatic number of the plane to 5 in 2018 using a graph that has >1000 nodes. "The chromatic number of the plane is …

The Hadwiger–Nelson problem asks: How many colors are required to color the plane such that no two points at distance 1 from each other have the same color?

Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their heights, then 50% of the curves have rank 0 and 50% have rank 1. See p. 28 of …

Original Hall's conjecture with exponent $1/2$.

There are no indecomposable vector bundles of rank 2 on $\mathbb{P}^n$ for $n \ge 7$. This is conjecture 6.3 in _VARIETIES OF SMALL CODIMENSION IN PROJECTIVE SPACE_, R. Hartshorne

Every convex set in $\mathbb R^2$ has VC dimension at most 3.

If $n \ge 2$, every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most 1.

Every convex set in $\mathbb R^3$ has $\mathrm{VC}_2$ dimension at most 1.

Is there a Lindelöf space with singletons as Gδ sets with cardinality greater than the continuum?

Let $G$ be a group, and let $A = \{a_1G_1, \dots, a_kG_k\}$ be a finite system of left cosets of subgroups $G_1, \dots, G_k$ of $G$. Herzog and Schönheim conjectured that if …

The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20], who prove $$ f(n) ≤ 2^{n+O(\sqrt{n\log n})}. $$ [HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János …

Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact space contains a non-trivial convergent sequence?

Problem 14 in [Ar2013]: Is it possible to represent an arbitrary compact space as an image of a homogeneous compact space under a continuous mapping?

**The idempotent conjecture** If `G` is torsion-free, then `K[G]` has no non-trivial idempotents.

**Conjecture 1.6 (Even case).** For a nonempty isolate-free graph $G$ on $n$ vertices, if $D$ is even, then $(D + 2)^2 \cdot i(G) \leq (D^2 + 4) \cdot n$.

**Conjecture 1.6 (Odd case).** For a nonempty isolate-free graph $G$ on $n$ vertices, if $D$ is odd, then $(D + 1)(D + 3) \cdot i(G) \leq (D^2 + 3) \cdot n$.

Johnson's bound for the independence number of the Johnson graph.

**The infinitude of cousin primes** There are infinitely many primes $p$ such that $p + 4$ is prime.

Are there infinitely many composite Fermat numbers?

Are there infinitely many Fermat primes?

There are infinitely many $\zeta(2n + 1)$, $n \in \mathbb{N}$, that are irrational. [Ri00] Rivoal, T. (2000). _La fonction zeta de Riemann prend une infinité de valeurs irrationnelles aux entiers impairs_. Comptes …

A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \equiv 1 \pmod{p^2}$, where $L_p$ is the $p$-th Lucas number. It is conjectured that there are infinitely many Wall-Sun-Sun primes.

A Lucas–Wieferich prime associated with $(a,b)$ is an odd prime $p$, not dividing $a^2 - b$, such that $U_{p-\varepsilon}(a,b) \equiv 0 \pmod{p^2}$ where $U(a,b)$ is the Lucas sequence of the first kind …

**Infinitude of betrothed numbers conjecture.** Are there infinitely many betrothed number pairs?

**Infinitely many even perfect numbers conjecture.** Are there infinitely many even perfect numbers? This is equivalent to asking whether there are infinitely many Mersenne primes, since by the Euclid–Euler theorem an even …

**Conjecture:** Are there infinitely many Leinster groups? This asks whether there exist infinitely many (non-isomorphic) finite groups that are Leinster groups. Formalized via the negation of "Does there exist an n such …

Are there infinitely many Mersenne primes?

**Infinitely many perfect numbers conjecture.** Are there infinitely many perfect numbers? *Reference:* [Wikipedia](https://en.wikipedia.org/wiki/Perfect_number)

There are infinitely many prime numbers of the form `k * 2 ^ k - 1` for `k > 1`.

There are infinitely many prime Pell numbers

It is not known whether there is an inifinite number of prime Euclid numbers.

Are there infinitely many primes $p$ such that $p - 1$ is a perfect square? In other words: Are there infinitely many primes of the form $n^2 + 1$?