10 problems

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…

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…

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 $G$ 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…

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

Let $\Delta_{n,\ell}$ be the simplicial complex of $\ell$-zero-sumfree subsets. Regard it as connected when for any two vertices there is a sequence of vertices joining them throug…

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 $(G,+)$ 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 suppose that … A simple ordering of $A$…

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…