5 problems

Let $K_n^{(k)}$ denote the complete $k$-uniform hypergraph, and let its edges be coloured with $r$ colours. A monochromatic tight-cycle partition is a partition of the vertex set i…

Let $B^n$ be the unit ball in $n$-dimensional Euclidean space, partitioned into $n+1$ subsets, and let $d$ denote the maximum diameter of the pieces. The standard simplicial decomp…

Two-dimensional box partition conjecture.

For $inmathbb{N}$ and $mathbf{x}inmathbb{N}^{n}$, a zero partition of $mathbf{x}$ is a choice of signs $sigmain{-1,1}^{n}$ such that $langlemathbf{x},sigmarangle=0$. **Zero-pa…

Let $K$ be a convex body in $\mathbb{R}^d$, let $\mu$ be an absolutely continuous probability measure on $\mathbb{R}^d$, let $n>1$ be a natural number, and let…