23 problems

Let $r,d$ be positive integers, and let a Tverberg partition of a set of points be a partition into $r$ parts whose convex hulls intersect. For a set of $(r-1)(d+1)+1$ points in…

Let $Q_1,\dots,Q_{d+1}$ be pairwise disjoint sets of $k$ points each in $\mathbb{R}^d$. A transversal is a subset of their union containing exactly one element from each $Q_i$. Col…

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 $d$ 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 $d$ 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 $K_{3(q-1)+1}$ 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…

Let $d\geq m\geq 1$, let $k$ be a positive integer, and let $F_1,F_2,\dots,F_m$ be families of convex sets in $\mathbb{R}^d$, with … which satisfy the colorful intersection propert…

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…

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, and let $(d_1,\dots,d_r)$ be Tverberg admissible, meaning that $\lfloor d/2\rfloor\leq d_i\leq d$ and $\sum_i d_i=(r-1)d$. A finite point s…

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 $m$ and $d$ be integers with $0\leq m\leq d-1$, let $r_0,\ldots,r_m\geq 1$ be integers, and set … For sets $X_0,\ldots,X_m\subset\mathbb{R}^d$ with…

The colored Tverberg problem. Determine the smallest natural number $n=n(d,r)$ for which every such collection admits this partition; the conjecture is that

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 $d\geq 2$, let $r$ and $k$ be positive integers, and let $F_1,F_2,\ldots,F_{(r+1)d+1}$ be colour classes, each consisting of $k$ points in $\mathbb{R}^d$. A colourful $k$-parti…

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…

Let $G$ be a graph on $(d+1)(q-1)+1$ vertices, with maximal degree $[?]$ unavailable

Let $t(d,r)$ denote the smallest integer $t$ such that, for every set $C\subset\mathbb{R}^d$ of $(d+1)t$ points partitioned into $t$-point subsets $C_1,\ldots,C_{d+1}$, there is a…

The colorful Tverberg theorem concerns collections of colored points and partitions satisfying the colorful condition that each partition block contains one point of each color, wi…