14 problems

Let $\mathbb{F}$ be a field, let $n\geq 3$, and let $\mathfrak{L}$ be a finite collection of $L$ lines in $\mathbb{F}^n$. For each joint $x\in J(\mathfrak{L})$, let $N(x)$ be the n…

For positive integers $k_1,\ldots,k_r,m_1,\ldots,m_r$, let $C_{k_1,\ldots,k_r;m_1,\ldots,m_r}$ be the optimal constant in the corresponding joints problem, and let…

Let $N$ be a positive integer, and consider $N$ two-dimensional affine flats in $\mathbb{F}^6$. A joint is a point incident to sufficiently many of these flats with linearly indepe…

Let $\mathbb{F}$ be a field, and let $(\mathcal{J},\mathcal{L})$ be a joints configuration in $\mathbb{F}^d$, with $J=|\mathcal{J}|$ joints and $L=|\mathcal{L}|$ lines. Assume that…

Let $(\mathbf{J};\mathbf{F}_1,\ldots,\mathbf{F}_r)$ be a joint set system arising from a generically induced joints configuration, and let the corresponding joints problem have its…

Let $k$ hyperplanes in $\mathbb{R}^d$ be in general position, and consider the $(d-1)$-wise intersections of the hyperplanes, giving $N=\binom{k}{d-1}$ lines. The $d$-wise intersec…

In the generalized setting, let $\bm d=(d_1^{m_1},\dots,d_r^{m_r})$ and consider a $\bm d$-joints configuration in $\mathbb F^d$. A generically induced configuration is formed by c…

Let $d_1,\dots,d_r$ and $m_1,\dots,m_r$ be positive integers, put $d=d_1m_1+\cdots+d_rm_r$, and let a $(d_1^{m_1},\dots,d_r^{m_r})$-joints configuration…

For a multijoints configuration $(\mathcal J,\mathcal L_1,\mathcal L_2,\mathcal L_3)$ in $\mathbb F^3$, let $|\mathcal L_i|$ denote the size of the $i$th line family. The multijoin…

Let $(\mathcal J,\mathcal L_1,\dots,\mathcal L_d)$ be a multijoints configuration in $\mathbb F^d$. A configuration is generically induced if the lines arise as $(d-1)$-wise inters…

A joints configuration consists of points and lines (or the corresponding higher-dimensional variants) satisfying the incidence conditions of the joints, multijoints, or higher-dim…

Let $N$ lines be given in ${\mathbb R}^3$, and call a point a joint if at least three non-coplanar lines pass through it. The joints conjecture. The number of joints is at most…

Let $\mathfrak{L}_1,\ldots,\mathfrak{L}_n$ be transversal collections of $L_1,\ldots,L_n$ lines in $\mathbb{F}^n$, meaning that whenever $n$ lines, one from each collection, meet a…

Multiplicity joints conjecture. The number of joints counted with multiplicities satisfies