69 problems

Classification conjecture. The only possibilities for $\phi$ are

Let $p$ be a prime, let $\mathbb{Z}_p$ be the cyclic group of order $p$, and let $\mathcal A(\mathbb{Z}_p)$ denote the set of minimal zero-sum sequences over this group. A set…

Let $n$ be a positive integer, let $T$ be a sequence of length $n$ over the cyclic group $\mathbb{Z}_n$, and call a subsequence of $T$ index one when its index is one. Lemke-Kleitm…

Let $G$ be the cyclic group of order $n$, and let $S$ be a minimal zero-sum sequence over $G$ of length $4$. The index $\operatorname{ind}(S)$ is the smallest number in … where…

For an integer $n\geqslant2$, let $\rho(n)$ denote the smallest prime number that does not divide $n$. For the cyclic group $\mathbb{Z}_{\!/n}$, let $\lambda(\mathbb{Z}_{\!/n})$ de…

Let $p$ be a prime, and let $A\subseteq\mathbb{Z}_p\setminus\{0\}$. An ordering $a_1,\ldots,a_{|A|}$ of $A$ is valid if its partial sums … are pairwise distinct. Graham's conjectur…

Let $p$) be prime and let $a_1,a_2,\dots,a_d$ be distinct nonzero elements of $\mathbb{Z}_p$. A set of elements is rearrangeable if its elements can be ordered so that all partial…

The high-density $3k-4$ conjecture. There are arithmetic progressions $P_A,P_B,P_C\subseteq G$, all having a common difference, such that

Let $\mathbb{Z}_N$ be the cyclic group of order $N$, and let $A\subset\mathbb{Z}_N$ be $\{x,2x,\ldots,(d-1)x\}$-free, meaning that no $x$ has all of $x,2x,\ldots,(d-1)x$ in $A$. Cy…

Cyclic cube-free density conjecture. If $d\mid N$, then

Generalized Maximum Modulus Principle. There exists $C(d,N)>0$, independent of $n$, such that for every $n\geq 1$,

Divisibility conjecture for n-circular bayonet codes. For any $n$-cbc $X$ and any $d$ prime to $n$, the set $\varphi_d(X)$ is an $n$-cbc.

Bajnok's limit conjecture. The value of

Let $T\subseteq \mathbb{Z}_v\setminus\{0\}$. An ordering of the elements of $T$ is simple when its partial sums are all distinct and non-zero. Graham–Archdeacon conjecture. The set…

Let $T\subseteq \mathbb{Z}_v\setminus\{0\}$ satisfy … An ordering of the elements of $T$ is simple when its partial sums are all distinct and non-zero. Alspach's conjecture. Every…

Let $\mathbb{Z}_m$ be a cyclic group of order $m=2^n$. Let $t$ and $m_1,\ldots,m_t$ be positive integers satisfying … with $m_1\geq 1$ and $m_i\geq 2$ for $2\leq i\leq t$. Equal-su…

Fix positive integers $r_1,\ldots,r_k$, and let $p_1,\ldots,p_k$ be distinct primes. A saturated transfer system on a finite group is a transfer system satisfying the two-out-of-th…

Przytycki–Vojtěchovský conjecture.

Let $n$ be even and $k$ be odd with … Set … Write $\langle h\rangle$ for the subgroup generated by $h$ in the relevant cyclic group. Sharpness conjecture. If…

Line-intersection bound conjecture. For every $x\in\mathbb{Z}_M$,

Fiberedness conjecture. For every $i$ there exists $1\leq\alpha_i<n_i$ such that either $A$ or $B$ is $M/p_i^{\alpha_i}$-fibered in the $p_i$ direction. In particular, if $M$ has…

Three-prime-factor reduction conjecture. The answers to Questions $q$-slab (both versions) and $q$-obstruction are affirmative when $M$ has at most $3$ distinct prime factors.

Let $n$ be a positive integer, let $\mathbb{Z}_n$ be the cyclic group of order $n$, and let $A\subseteq\mathbb{Z}_n$. For a subset $S\subseteq\mathbb{Z}_n$, write $2S=S+S$. Small-d…

Let $m$, $n$, and $k$ be positive integers with $m\geq 2$, and suppose … For a prime $p$ dividing $|G|$, write $\operatorname{ord}_p(m)$ for the multiplicative order of $m$ modulo…

Let $k$, $\omega$, and $n$ be positive integers, and let $\mathbb{Z}_{\omega}$ be a finite cyclic group of order $\omega$. Suppose … A subset $A$ is a direct factor of…