21 problems

Let $d$ be a positive integer. For a finite family of convex sets in $\mathbb{R}^d$, suppose that the intersection of every $2d$ members has volume at least one. The Quantitative V…

Let $cmathcal F$ be a finite family of convex sets in $cmathbb R^d$ such that the intersection of every $2d$ or fewer members has volume at least $1$. The conjecture asserts that ……

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 $W$ be a finite set and let $X$ be a $d$-Leray simplicial complex with vertex set $W$, meaning that the reduced homology groups of every induced subcomplex $Y$ vanish in all di…

Helly–Radon equality conjecture.

Given a natural number $k$ and an object $B$, consider collections of $k$ translated copies of $B$ in the property-testing setting for $n$ points. Fractional Helly-type 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 $\mathcal{F}$ be an infinite family of convex closed sets in the plane, with at least one bounded member. A $(4,3)$-theorem means that if every four members of $\mathcal{F}$ co…

For a positive integer $d$ and $v\in(0,1)$, let $\mathcal{Q}_d(v^\ell)$ denote the quantitative hypergraph chain whose level-$\ell$ hyperedges correspond to subfamilies of convex s…

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…

Bárány–Katchalski–Pach conjecture. Under these hypotheses, the intersection of the entire family has diameter at least $c d^{-1/2}$.

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 $\{P_i:i\in I\}$ be a family of closed convex sets in $\mathbb{R}^n$ such that … The authors ask whether there exist $s\leqslant 2n$ and $i_1,\ldots,i_s\in I$ such that … where…

Let $d>1$ and let $\beta(\alpha,d)$ denote the smallest value such that, whenever finite color classes of convex sets in $\mathbb{R}^d$ have at least an $\alpha$ proportion of thei…

Let $G$ be an additive subgroup of $\mathbb{R}^d$, and let $h(G)$ denote its Helly number. Finite subgroup Helly conjecture. For every subgroup $G\subset\mathbb{R}^d$, the Helly nu…

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…

Let $\mathcal{H}=\{A_1,\ldots,A_n\}$ be a family of pairwise disjoint convex sets in $\mathbb{R}^d$. For each $i$, let $T_k(A_i)$ denote the set of $k$-dimensional affine subspaces…