5 problems

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 $\mathbb{F}$ 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 $d_1,\dots,d_r$ 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…

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