6 problems

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…

Let $d\beq 1$, $r\geq d+1$, $1\leq m\leq r$ and $m\leq k\leq \min\{m+d,r\}$ be integers. Let $V$ be a finite set with $|V|\geq \max\{m+d,r\}$. Let $X$ be a $d$-Leray simplicial com…

A simplicial complex $\mathsf{K}$ is $d$-Leray if, for every induced subcomplex $\mathsf{L}\leq\mathsf{K}$ and every $i\ge d$, its reduced homology group…

Let $d$ be a non-negative integer. For each $i\in[d+1]$, let $n_i$ be positive and let $r_i$ be a non-negative integer with $n_i\ge r_i+1$. Let…

Let $\beta_{\operatorname{col}}(\alpha,d)$ denote the largest proportion guaranteed in the colorful fractional Helly theorem for $d+1$ finite nonempty families of convex sets in…

Transversal-dimension conjecture. For every integer $k$ with $1\leq k\leq d$, there exist numbers $h(k,d)$ such that, for every $d$-colored family $\mathcal{F}$ satisfying…