16 problems

Let $d$ and $q$ be positive integers. For a continuous map $f:S^{d-1}\to\mathbb{R}^d$ and a point $p\notin f(S^{d-1})$, define the winding number $W(f,p)\in\mathbb{Z}$ by the homol…

Let $m(d,r)$ denote the minimum number of unordered Tverberg partitions of a set of $(d+1)(r-1)+1$ points in $\mathbb{R}^d$, where a Tverberg partition of order $r$ is a partition…

Let $r$ and $d$ be positive integers, and let $N$ denote the number of points under consideration. Birch's conjecture. Any $(r-1)(d+1)+1$ points in $\mathbb{R}^{d}$ can be partitio…

Let $F_1,\ldots,F_{d+1}$ be families of $r$ points each in $\mathds{R}^d$, considered as color classes. A colorful partition is a partition into $r$ sets $A_1,\ldots,A_r$ such that…

Let $d$ and $q$ be positive integers, and let $\Delta_{(d+1)(q-1)}^{\le d}$ denote the $d$-skeleton of the simplex. A Tverberg partition is a collection of $q$ disjoint faces whose…

Let $K_{3(q-1)+1}$ be the complete graph on $3(q-1)+1$ vertices, and consider any drawing of this graph in the plane. For a triangle of edges and a point not on it, its winding num…

Let $0\leq k\leq d-1$ and let $S_0,S_1,\ldots,S_k$ be finite sets of points in $\mathbb R^d$, with … for $i=0,1,\ldots,k$. A common convex-hull transversal is a $k$-flat meeting ea…

The bounded-excess connectivity conjecture. There exists a constant $k(d)$ depending only on $d$ such that $G_T[S,r]$ is connected whenever

Let $S$ be a set of $n$ points in $\mathds{R}^d$ in strong general position, and let $G_T[S,r]$ be its Tverberg $r$-partition graph. Write $\delta(G_T[S,r])$ for the minimum degree…

Let $q \ge 2$ and $d \ge 1$ be integers. Let $C_1, \ldots, C_{d+1}$ be pairwise disjoint sets of vertices of $\Delta_{q(d+1)-1}$, each of size $q$, and let…

Let $T(d,r,k)$ be the least positive integer $n$ such that every set of $n$ points in $\mathbb{R}^d$ admits a partition into $r$ disjoint parts whose convex hulls have nonempty int…

Fix integers $r\geq 2$ and $d\geq 1$. A partition $X_1\sqcup\cdots\sqcup X_r$ of $\{1,2,\dots,(r-1)(d+1)+1\}$ is colorful if, for every $1\leq k\leq d+1$, the block … contains exac…

Let $r\geq 2$ and $d\geq 1$ be integers. An $r$-tuple $(d_1,\dots,d_r)$ is Tverberg admissible if … and $\sum_i d_i=(r-1)d$. AP conjecture. Every Tverberg admissible $r$-tuple is T…

Let $r,d$ be fixed positive integers. A colorful partition of $M$ families of $r$ points each in $\mathds{R}^d$ is a partition into $r$ sets $A_1,\ldots,A_r$ assigning one point fr…

Let ${\mathcal B}^d=[0,1]^d$ be the $d$-dimensional box equipped with max-T convexity, and let $X$ be a set of $(d+1)(r-1)+1$ points in ${\mathcal B}^d$. Tverberg's theorem for max…

Let $F_1, F_2, \ldots, F_{d+1}$ be colour classes, each consisting of $k$ points in $\mathbb{R}^d$. A colourful $k$-partition is a partition of their union into $k$ pairwise disjoi…