17 problems

Let $q\geqslant 2$ and let $\mathrm{KG}^q(n,q)$ be the $q$-uniform Kneser hypergraph on the $q$-subsets of $[n]$, with edges consisting of $q$ pairwise disjoint vertices. Let…

For positive integers $q,n,k$ with $n\ge qk$, let $\mathrm{KG}^q(n,k)$ be the $q$-uniform Kneser hypergraph whose vertices are the $k$-element subsets of $\{1,2,\dots,n\}$ and whos…

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 $r\geq 3$, let $[n]=\{1,\ldots,n\}$, and let $\mathcal G$ be a hypergraph over the ground set $[n]$. Write $\mathcal G_{r\text{-stable}}$ for the hypergraph obtained by restric…

For $n\geq rk$, let $KG^r(k,n)$ be the hypergraph whose vertices are the $k$-subsets of $[n]=\{1,2,\ldots,n\}$ and whose edges are the $r$-tuples of pairwise disjoint $k$-sets. A s…

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 $n,k,r$ be integers with $n\geq rk$ and $r>2$, and let $\operatorname{KG}^r_2(n,k)$ be the $r$-uniform hypergraph whose vertices are the $k$-subsets of $\{1,\dots,n\}$ and whos…

For positive integers $q,n,k$ with $n\ge qk$, let $\mathrm{KG}^q(n,k)_{\widetilde{q\text{-stab}}}$ be the Kneser hypergraph whose vertices are the $k$-element subsets of…

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 $n$, $k$, and $r$ be positive integers with $k\geq 2$, $r\geq 3$, and $n>r\left(k+\frac12\right)$. Define … Here $\operatorname{KG}^{r}_{n,k}(p)$ is the random $r$-uniform Knes…

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…

Let $r\ge 2$, $k\ge 1$, and $n\ge sk$, and consider colorings of $\binom{[n]}{k}$ in which any $r$ same-colored members contain some $s$ members with a common element. Lange–Ziegle…

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…

Let $n$, $k$, and $r$ be positive integers, and let $\operatorname{KG}^r(n,k)$ be the $r$-uniform Kneser hypergraph whose vertices are the $k$-subsets of $[n]$, with adjacency give…

Erdős's conjecture. This bound is sharp for all $r\ge 2$, that is,