6 problems

Let $n$ and $d$ be positive integers, and let $\mathcal{F}\subseteq\binom{[n]}{d+1}$ be a $(d+1)$-uniform family with VC-dimension at most $d$. Erdős–Frankl–Pach conjecture. When…

Let $0\le s\le d$, and let $\mathcal{F}\subseteq\binom{[n]}{d+1}$ be an $s$-witness family, meaning that for every $F\in\mathcal{F}$ there exists $B_F\in\binom{F}{s}$ such that…

Let $0\le s\le d$, and let $\mathcal{F}\subseteq\binom{[n]}{d+1}$ be an $s$-witness family if for every $F\in\mathcal{F}$ there exists $B_F\in\binom{F}{s}$ such that…

Chao–Xu–Yip–Zhang's conjecture.

O'Neill and Versträete's conjecture. Let $k>r\geq2$ and $n\geq kr(r-1)$. Then the unique extremal non-trivial $r$-wise intersecting families in $\binom{[n]}k$ are ${\rm BD}_r(n,k)$…

For $n\in\mathbb{N}$ and $r,t,i,j\in\{0\}\cup[n]$ with $1\leq t\leq r$ and $t+i+j\leq n$, define … Let $m,n\in\mathbb{N}$, $r\in[m]$, $s\in[n]$, $t\in[\min\{r,s\}]$,…