Unsolved Problems

Conjecture The largest measure of a Lebesgue measurable subset of the unit sphere of $\mathbb{R}^n$ containing no pair of orthogonal vectors is attain...

Question What is the saturation number of cycles of length $2\ell$ in the $d$-dimensional hypercube?...

Problem Determine the smallest percolating set for the $4$-neighbour bootstrap process in the hypercube....

Problem Bound the extremal number of $C_{10}$ in the hypercube....

Conjecture Does there exist a nontrivial perfect 2-error-correcting code over any finite alphabet, other than the ternary Golay code?...

A $(v, k, t)$ covering design, or covering, is a family of $k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is contained i...

Conjecture If $A$ is an invertible $n \times n$ matrix, then there is an $n \times n$ submatrix $B$ of $[A A]$ so that $perm(B)$ is nonzero....

Conjecture If $B_1,B_2,\ldots B_p$ are invertible $n \times n$ matrices with entries in ${\mathbb Z}_p$ for a prime $p$, then there is a $n \times (p-...

Question Is the binary affine cube $AG(3,2)$ the only 3-connected matroid for which equality holds in the bound $$E(M) \leq c(M) c(M^*) / 2$$where$c(M...

Question Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\geq n_0(k)$, then every saturated $k$-Sperner system $\mathcal{...

For $k$ and $\ell$ positive integers, the (mixed) van der Waerden number $w(k,\ell)$ is the least positive integer $n$ such that every (red-blue)-colo...

We let $Q_d$ denote the $d$-dimensional cube graph. A map $\phi: E(Q_d) \rightarrow \{0,1\}$ is called edge-antipodal if $\phi(e) \neq \phi(e')$ whene...