Unsolved Problems

For $k\in \mathbb{N}$ let $T_k$ denote the minimal number $t\in \mathbb{N}$ such that there is a rainbow $AP(k)$ in every equinumerous $t$-coloring of...

Problem Do 4-colorings of $\mathbb{Z}_{p}$, for $p$ a large prime, always contain a rainbow $AP(4)$ if each of the color classes is of size of either ...

Question Is it true that every permutation of positive integers must contain monotone 4-term arithmetic progressions?...

Question Is the set of prime numbers 2-accessible?...

Question Is the set of Fibonacci numbers 3-accessible?...

Conjecture An integer partition is wide if and only if it is Latin....

Begin with the generating function for unrestricted partitions: (1+x+x^2+...)(1+x^2+x^4+...)(1+x^3+x^6+...)... Now change some of the plus signs to ...

Conjecture Define a $2 \times n$ array of positive integers where the first row consists of some distinct positive integers arranged in increasing ord...

Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \times n$ grid. The first player (if...

Problem Two players alternately write O's (first player) and X's (second player) in the unoccupied cells of an $n \times n$ grid. The first player (if...

Conjecture Every surreal number has a unique sign expansion, i.e. function $f: o\rightarrow \{-, +\}$, where $o$ is some ordinal. This $o$ is the leng...

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...