10 problems

Let $n,s$ be integers with $n\geq 2s-1$, let $L=\{\ell_{1},\ell_{2},\ldots,\ell_{s}\}$ be a set of $s$ non-negative integers, and let an $L$-intersecting Sperner system be a Sperne…

Let $L=\{\ell_{1},\ell_{2},\ldots,\ell_{s}\}$ be a set of $s$ non-negative integers and $K=\{k_{1},k_{2},\ldots,k_{r}\}$ be a set of $r$ positive integers with … Let an $L$-interse…

Frankl's conjecture. There exists a constant $c$ such that, whenever $k>c d^2$,

Ellis–Friedgut–Pilpel conjecture. For any $n\in\mathbb N$ and $1\leq t\leq n$, the maximum size of a $t$-intersecting family in $S_n$ is attained by one of the families…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. Let $Y,Z\subseteq G_n$ be $t$-space-cross-intersecting sets, meaning that every pair in $Y\times Z$ is $t$-space-inte…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. A subset $Y\subseteq G_n$ is $t$-space-intersecting when every pair of elements of $Y$ is $t$-space-intersecting. For…

Let $q$ be a prime power and let $G_n=\operatorname{GL}(n,q)$. For positive integers $t$ and sufficiently large $n$ compared to $t$, let $Y\subseteq G_n$ be a $t$-intersecting set,…

Ellis's conjecture. If $\mathcal{F}$ is a $t$-setwise intersecting family of $\operatorname{sym}(n)$, then

Gerbner et al.'s conjecture. Then

Let $0<s\leq k\leq n$ be positive integers. Let $L=\{\ell_1,\ldots,\ell_s\}$ be a set of $s$ positive integers such that $0<\ell_1<\cdots<\ell_s$. Suppose that $n>k^2-k+1$. Let…