7 problems

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…

Let $r,d$ be positive integers, and let $A_1,\dots,A_{d+1}$ be sets of $r$ hyperplanes each in $\mathbb{R}^d$. A partition of their union into $r$ sets $B_1,\dots,B_r$ is colorful…

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…