10 problems

Helly–Radon equality conjecture.

Let $\mathcal{F}_1, \dots, \mathcal{F}_{2d}$ be finite families of convex sets in $\mathbb{R}^d$. Assume that for any choice…

Macbeath point conjecture. The inclusion

Let $\mathcal{P}$ be the set of prime numbers, and let $h(S)$ denote the Helly number of a set $S\subseteq\mathbb{R}^2$. De Loera–La Haye–Oliveros–Roldán-Pensado conjecture. … This…

Let $\mathcal{F}$ be an intersecting family of convex sets in $\mathbb{R}^3$. For each triple $T=\{F_1,F_2,F_3\}\in {\mathcal{F}\choose 3}$, let $D_T\subset S^2$ consist of the dir…

Colorful Bárány–Katchalski–Pach conjecture. Under these hypotheses, there is an index $k\in[2d]$ whose entire family has intersection of diameter at least $c d^{-1/2}$, for some ab…

Let $p$ and $q$ be integers with $p\geq q\geq 3$, let $S$ be a surface, and let $\mathcal{F}$ be a finite family of open connected subsets of $S$. The $(p,q)$ property means that a…

Let $\mathcal{A}$ be a finite family of linear operators $\mathbb K^d\to\mathbb K^d$, where $\mathbb K$ is an arbitrary field and $d\geqslant 2$. Define $HI(\mathbb K^d)$ to be the…

Let $\mathbf{K}$ be a convex body in $\mathbb{E}^d$, $d\geq 3$, and let $\operatorname{him}(\mathbf{K})$ be its Helly dimension, defined as the least $h$ for which the correspondin…

Let $h(S)$ denote the Helly number of a subset $S\subseteq\mathbb{R}^d$, and let $\mathbb{P}$ be the set of prime numbers. Prime-square Helly conjecture. … This conjecture predicts…