Unsolved Problems

A graph $G$ is $k$-degenerate if every subgraph of $G$ has a vertex of degree $\le k$. Conjecture Every simple planar graph has a 5-coloring so that ...

The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane. Conjecture $\displaystyle cr(K_n) = \frac ...

The crossing number $cr(G)$ of $G$ is the minimum number of crossings in all drawings of $G$ in the plane. Conjecture $\displaystyle cr(K_{m,n}) = \f...

We let $cr(G)$ denote the crossing number of a graph $G$. Conjecture Every graph $G$ with $\chi(G) \ge t$ satisfies $cr(G) \ge cr(K_t)$....

Problem Does the following equality hold for every graph $G$? $$ \text{pair-cr}(G) = \text{cr}(G) $$ The crossing number $\text{cr}(G)$ of a graph $G...

We say that a set $P \subseteq {\mathbb R}^2$ is $n$-universal if every $n$ vertex planar graph can be drawn in the plane so that each vertex maps to ...

Conjecture Is there a graph $G$ that is a minor-minimal obstruction for two non-orientable surfaces?...

Problem Does there exist a finitely presented group of intermediate growth?...

Problem Find a constant $k$ such that for any $d$ there is a sequence of tautologies of depth $k$ that have polynomial (or quasi-polynomial) size proo...

Conjecture Suppose $k$ runners having distinct constant speeds start at a common point and run laps on a circular track with circumference 1. Then for...

Problem Let $A \subseteq {\mathbb N}$ be a set of $n$ positive integers and set $$ m(A) = - \min_x \sum_{a \in A} \cos(ax). $$ What is $m(n) = \min_A...

Call a polynomial $p \in {\mathbb Q}[x]$ rationally derived if all roots of $p$ and the nonzero derivatives of $p$ are rational. Conjecture There doe...

Conjecture $\pi$ and $e$ are algebraically independent...

Question Is Euler-Mascheroni constant an transcendental number?...

Conjecture Are all Fermat Numbers $$ F_n = 2^{2^{n } } + 1 $$ Square-Free?...

Conjecture A Fermat prime is a Fermat number $$ F_n = 2^{2^n } + 1 $$ that is prime. The only known Fermat primes are F_0 =3,F_1=5,F_2=17,F_3 =257,F_...

Conjecture Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free?...

Say that a set $S \subseteq {\mathbb Z}$ has distinct subset sums if distinct subsets of $S$ have distinct sums. Conjecture There exists a fixed cons...

Conjecture There is no covering system whose moduli are odd, distinct, and greater than 1....

For a finite (additive) abelian group $G$, the Davenport constant of $G$, denoted $s(G)$, is the smallest integer $t$ so that every sequence of elemen...

Conjecture Let $G$ be an abelian group of odd order and let $A,B \subseteq G$ satisfy $|A| = |B| = k$. Then the elements of $A$ and $B$ may be ordered...

Conjecture Formulate a central limit theorem for the KPZ universality class....

Problem Given two codes $R,C$, their Tensor Product $R \otimes C$ is the code that consists of the matrices whose rows are codewords of $R$ and whose ...

Problem Let $a_1,a_2,\ldots,a_n$ be natural numbers with $\sum_{i=1}^n a_i < 2^n - 1$. It follows from the pigeon-hole principle that there exist dist...

If $p$ is prime and $g,h \in {\mathbb Z}_p^*$, we write $\log_g(h) = n$ if $n \in {\mathbb Z}$ satisfies $g^n = h$. The problem of finding such an int...

Problem Prove unconditionally that $\mathcal{AM}$ $\subseteq$ $\Sigma_2$....

Conjecture Can all problems that can be computed by a probabilistic Turing machine (with error probability < 1/3) in polynomial time be solved by a de...

Conjecture For every rational $\epsilon > 0$ and every rational $\Delta$, there is no polynomial-time algorithm for the following problem. Given is a...

Question Is there a hyperbolic 3-manifold whose fundamental group rank is strictly less than its Heegaard genus? How much can the two differ by?...

Problem Determine a computable set of invariants that allow one to determine, given a compact boundaryless 3-manifold, whether or not it embeds smooth...

Problem Is there an algorithm which takes as input a triangulated 4-manifold, and determines whether or not this manifold is combinatorially equivalen...

Problem Does there exist a smooth/PL embedding of $S^2$ in $S^4$ such that the fundamental group of the complement has an unsolvable word problem?...

Problem Let $\mathcal{X}$ be an indexed family of filters on sets. Which of the below items are always pairwise equal? 1. The funcoid corresponding t...

Problem Two players start at a distance of 2 on an (undirected) line (so, neither player knows the direction of the other) and both move at a maximum ...