32 problems

Kalai–Meagher conjecture. For every $n\geq 3$,

Let $f_\ell(n,k)$ denote the clique number of the xor-product of $\ell$ copies of the Kneser graph $KG(n,k)$. Fix $\ell$ and suppose that $k$ is sufficiently large. Higher-power gr…

Let $n,k$ be positive integers with $k\geq2$, and let $\vec{s}=(s_1,\ldots,s_k)$ be a positive integer vector with $s_i\geq2$ for $1\leq i\leq k-1$. Let…

Exact homology-rank conjecture. The rank of the $p$-dimensional homology group of $\operatorname{Ind}(\operatorname{KG}(n,k))$ is

Dense minimizer conjecture. There exists a family

Sparse minimizer conjecture. There exists a minimizer $\mathcal{F}$ such that

Let $q$ be a prime power with $q\equiv 3\pmod{4}$, and consider the Kneser graph $K(q+1,3)$ and its automorphism group. Maximum-density conjecture. The group…

Let $q$ be an odd prime power, and let $\operatorname{PGL}_2(q)$ act on the $2$-sets of $\{1,\dots,q+1\}$. Intersection-density conjecture. The action has intersection density $1$.…

Let $q$ be a prime power, and let $\operatorname{PSL}_2(q)$ act on the $3$-sets of its natural degree-$q+1$ action. Intersection-density conjecture. If $q^2\equiv 1\pmod{5}$, then…

Let $d\ge 2$ be an integer, and let $q\Gamma_{2d+1,\{d,d+1\}}$ be the Kneser graph whose vertices are the relevant subspaces and whose adjacency is determined by dimensions $d$ and…

For integers $n$ and $k$, let $K_i(n,k)$ denote the random subgraph process of the Kneser graph, and define … … … … Here an EKR graph is one whose maximum independent sets are prec…

Let $KG(n,k)$ be the graph whose vertices are the $k$-subsets of $[n]=\{1,\ldots,n\}$, with two vertices adjacent exactly when the corresponding subsets are disjoint. A vertex cut…

Let $B=\{0,1\}^n$, let $\mu$ be the uniform measure on $B^t$, and let $p_{\mathrm{intersecting}}(t)$ be the limiting maximal measure obtained when each player's winning family is a…

The toughness conjecture for Kneser graphs. If $k\geq 5$ and $n\geq 2k+1$, then

Let $G$ be a graph on $N$ vertices. Suppose there is a parameter $W$ such that, for every induced subgraph $G'$ and every $n$, all but at most … edges of $G'$ are contained in some…

Let $m>w\ge 1$. The Kneser graph $\operatorname{KG}(m,w)$ has as vertices the $w$-element subsets of an $m$-element set, with two vertices adjacent when the corresponding subsets a…

Stahl's conjecture. For all integers $k$ and $d\geq 2s$,

For integers $n>k\geq 1$, let $\operatorname{KG}(n,k)$ be the graph whose vertices are the $k$-subsets of $[n]$, with two vertices adjacent exactly when the corresponding sets are…

Let $K(m,n)$ be the Kneser graph whose vertices are the $n$-subsets of $\{1,\ldots,m\}$, with two vertices adjacent when the corresponding subsets are disjoint. For an integer $r$,…

Let $KG(n,k)_{s-\mathrm{stab}}$ be the induced subgraph of the Kneser graph $KG(n,k)$ on the $s$-stable $k$-subsets of $[n]$, where a subset $A$ is $s$-stable when…

Let $\mathcal H\subset 2^{[n]}$ be a simplicial complex, and let $\mathcal H^{*2}_{\Delta}$ be its deleted join. Its $\mathbb Z_2$-index, denoted…

Let $KG_{n,k}$ be the Kneser graph on the $k$-element subsets of $[n]$, with edges joining disjoint sets. Consider a subset of its vertices that is properly colorable with $n-2k+1$…

Let $n,k,k'$ be integers, and write $k'=qk-r$ with $0\leq r<k$. Let $\operatorname{KG}(n,k)$ denote the Kneser graph on the $k$-subsets of $[n]$. Stahl's Kneser graph homomorphism…

Let $KG_{n,k}$ be the Kneser graph on the $k$-element subsets of $[n]$, and let $KG_{n,k}(p)$ be its random subgraph obtained by retaining each edge independently with probability…

For integers $n\geqslant s\geqslant2$ with $n>ks$, let $C_n^{s-1}$ be the $(s-1)$-st power of the cycle of length $n$, and let $K(n,k,s)$ be the $s$-uniform hypergraph whose vertic…