11 problems

For integers $d$, $r$, and $k$ with $2\le k\le r$, let $T(d,r,k)$ be the smallest integer $n$ such that every set of $n$ points in $\mathbb R^d$ can be partitioned into $r$ parts w…

Let $(X,\mathcal C)$ be an abstract convexity space with Radon number $D$, and let $\operatorname{Tv}_{\mathcal C}(r)$ denote its Tverberg number for $r$ parts. Eckhoff's partition…

Let $S\subset\mathds{R}^d$ be a set such that the Helly number $\mathds H_S(k)$ is finite for all $k$. For any $m,k$, there are integers $m_1$ and $m_2$. Given $m_1$ families…

Let $T(d,k)$ be a constant such that, for every set of $n$ points with $k$ independent and $d-k$ dependent degrees of freedom, there is a $k$-flat $F$ and a partition of the points…

Let $X$ be a Banach space, let $k\ge 2$ be fixed, and let $n\ge 2$. For $n$-point sets $Q_1,\dots,Q_n\subset X$, define $\operatorname{Tv}(X,k,n)$ as the smallest radius factor suc…

For integers $d,r\geq 1$, let $cmathcal F_1,\dots,\mathcal F_{2d}$ be families in $cmathbb R^d$, each containing $f(d,r)$ convex sets of volume $1$. A colorful Tverberg conjecture.…

Let $1\leq k\leq d$ be integers, and let $A_1,\dots,A_{d-k+1}$ be finite arrangements of hyperplanes in $\mathbb{R}^d$. Assume that … for positive integers $r_i$, for each…

For $r\geq 3$ and $d\geq r-1$, let $N_{(\Delta_{r-1};d)}$ denote the least number of vertices needed so that every map $f:\Delta_{N-1}\rightarrow\mathbb{R}^d$ admits a partition in…

Bárány–Kalai–Meshulam conjecture. There exists an integer $n_k\geq 1$ depending only on $k$ such that, whenever $M$ has at least $n_k$ disjoint bases, the complex $M^{*k}_\Delta$ i…

Let $3n$ straight lines in general position be given in the plane, and partition them into three color classes of $n$ lines each. The dual colorful Tverberg theorem. The lines can…

Let $\mathcal F$ be a family of $(d+1)n$ hyperplanes in general position in $\mathbb R^d$. A simplex formed by $d+1$ hyperplanes is the convex hull of the $d+1$ intersection points…