22 problems

Tannenbaum's zero-sum partition conjecture. For every positive integer $t$ and every integer partition $\{m_i\}_{i=1}^t$ of $m-1$, with $m_i\geq 2$ for every $i\in[1,|R|/2]$ and…

Let $N=\{n_1,n_2,\ldots,n_q\}$ be a set of $q$ natural numbers. There is a nonempty index set $I\subset\{1,\ldots,q\}$ such that $q\mid\sum_{i\in I}n_i$. Erdős–Lemke conjecture. Th…

Let $\Gamma$ be a finite Abelian group, and let $I(\Gamma)$ denote its set of involutions. A finite Abelian group has 4-ZSPP if every integer partition of $|\Gamma|-1$ whose parts…

Erdős–Lemke conjecture. Every such sequence contains a zero-sum subsequence whose sum is at most $\operatorname{lcm}(n,a_1,\ldots,a_n)$.

Cichacz's zero-sum partition conjecture. For every positive integer $t$ and every integer partition $\{m_i\}_{i=1}^t$ of $m-2$, with $m_i\geq 2$ for every $i\in[1,t]$, there is a s…

For a prime number $p$ and $d\in\mathbb{N}$, let $f(p,d)$ be the smallest positive integer such that, for every collection of $f(p,d)$ linear bases of the vector space…

Let $G$ be an abelian group and let $X\subseteq G\setminus\{0\}$. Call $X$ simply orderable when its elements can be arranged so that every nonempty proper consecutive subsequence…

Let $n=2M+1$ be a positive odd number, and let $d_1,d_2,\dots,d_M$ be elements of $\mathbb{Z}_n\setminus\{0\}$. Pasotti–Pellegrini conjecture. The set $\mathbb{Z}_n\setminus\{0\}$…

Erdős's conjecture. Every finite abelian group satisfies

Lemke–Kleitman conjecture. Every sequence of $|\Gamma|$ elements of a nonabelian finite group $\Gamma$ is good.

Skolem-partition characterization problem. Characterize all finite Abelian groups $\Gamma$ for which $R$ has a Skolem partition.

Row and column hypotheses. For all $n$, $\alpha_n^i$ is an increasing sequence for $0<i<n/2$, and for each $d$, $\alpha_i^d$ is an increasing sequence.

Let $t\geq 3$ be an integer, let $p\geq 3$ be a prime number, and let $w$ be a zero-sum-free $\mathbb{Z}_p$-arc-labelling of the complete digraph $\accentset{\leftrightarrow}{K}_t$…

Optimal discrepancy conjecture. This construction is best possible for every $n\geq9$; that is, every non-diagonal $n\times n$ zero-sum-square-free $\{-1,1\}$-matrix has discrepanc…

Linear discrepancy conjecture. For every $C>0$, there is an integer $N$ such that, whenever $n\geq N$, every non-diagonal $n\times n$ matrix $M$ with entries in $\{-1,1\}$ and

Zero-sum constant conjecture. One may take

Let $(G,+)$ be an abelian group. Let $A$ be a finite subset of $G\setminus\{0\}$ such that no $2$-subset $\{x,-x\}$ is contained in $A$ and … A simple ordering of $A$ is an orderin…

Let $\Delta_{n,\ell}$ be the simplicial complex of $\ell$-zero-sumfree subsets, and let $h_1,\ldots,h_d$ be the relevant entries of its $h$-vector. A simplicial complex is pure whe…

For an odd integer $n$, consider the complexes $\Delta_{n,(n-1)/2}$ and $\Delta_{n,(n+1)/2}$ of $\ell$-zero-sumfree subsets. A simplicial complex is pure when all of its facets hav…

Let $P$ be a polytope in $\mathbb{R}^n$ that contains the origin, and let its $1$-skeleton be the set of its vertices and edges. Bárány–Katchalski–Pach conjecture. The $1$-skeleton…

Dual cross-number conjecture. A zero-sumfree sequence whose cross number is at least the lower bound $\mathsf{k}^*(G)$ has length at most the corresponding maximal lower-bound leng…

Cross-number conjecture. Given a zero-sumfree sequence $S$ in $G$ with $\lvert S\rvert\geq\mathsf{d}^*(G)$, one has