11 problems

Let $KG^r(n,k)$ be the Kneser hypergraph whose vertices are the $k$-element subsets of $[n]$, and call a subset $S\subseteq[n]$ $r$-stable when any two elements $x,y\in S$ satisfy…

Let $[n]=\{1,\ldots,n\}$, and let $\mathrm{KG}^r(n,k)_{s\textup{-stab}}$ be the induced $r$-uniform Kneser hypergraph whose vertices are the $s$-stable $k$-subsets of $[n]$, where…

Let $[n]=\{1,\ldots,n\}$, let $\mathcal G$ be a hypergraph over the ground set $[n]$, and let $r\geq s\geq 2$. Write $\mathcal G_{\widetilde{\{s\text{-stable}\}}}$ for the correspo…

Let $r\geq 2$ and let $\mathcal{F}$ be a set system. Write $\mathcal{F}_{\widetilde{r-\operatorname{stab}}}$ for the family obtained by retaining the source's weakened notion of…

Let $n\ge 1$, $r\ge 2$, and $s\ge r$, and let $\cal F$ be a family of subsets of $[n]$. For the almost $s$-stable subfamily $\cal F_s$, let $\operatorname{KG}^r(\cal F_s)$ be the…

Let $q$ be a positive integer, and let $P$ be a path whose vertex set is partitioned into $m$ subsets $V_1,\dots,V_m$, each of size at least $q-1$. A set of vertices is $q$-stable…

Let $A\subset[n]$, and let $\mathrm{KG}^r_A(n,k)$ be the subhypergraph of the $r$-uniform Kneser hypergraph induced by the $k$-subsets $\sigma\subset[n]$ with $\sigma\not\subset A$…

Let $r\ge 2$, $k\ge 1$, and $n\ge rk$ be integers, and let $\mathcal P=\{P_1,\dots,P_\ell\}$ be a partition of $[n]$ such that $|P_i|\le r$ for every $i$. Let…

For positive integers $n,k,r$ with $k\geq 2$, $r\geq 2$, and $n\geq rk$, let $KG^{r}(n,k)_{r\text{-}stab}$ be the $r$-stable $r$-uniform Kneser hypergraph, and let $KG^{r}(n,k)$ be…

Ziegler–Alon–Drewnowski–Łuczak conjecture.

Let $\mathcal{G}=(V,E)$ and $\mathcal{G}'=(V',E')$ be hypergraphs. Their categorical product $\mathcal{G}\times\mathcal{G}'$ has vertex set $V\times V'$ and edges whose projections…