11 problems

Meunier's conjecture. For all $s\geq 2$ and $n\geq sk$,

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 $G$ be a B-graph, meaning that $sigma_v(G)=alpha(G)$, and suppose that $G$ has no isolated vertices. Let $omega_e(G)$ be the greatest natural number such that every edge of $G$…

Extremal-family conjecture. If $\mathcal{F}\subseteq\binom{[n]}{k}_{s\text{-stable}}$ is a non-star intersecting family with $\tau(\mathcal{F})\geq 2$, then, for $n$ large enough—s…

Let $G$ be the graph of a flag triangulation of the $(d-1)$-dimensional sphere on $n$ vertices, and let $\alpha_M(d,n)$ denote the maximum possible size of a stable set in such a g…

Let $G$ be the graph of a flag triangulation of the $(d-1)$-dimensional sphere on $n$ vertices, and let $I$ be a stable set of $G$. The half-density conjecture. For flag spheres, t…

Let $G$ be a graph whose clique complex is a flag triangulation of the $(d-1)$-dimensional sphere, and define … where $\alpha(G)$ is the size of a largest stable set of $G$. The ex…

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…

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…

Fair stable-set splitting conjecture. There always exist pairwise disjoint $q$-stable sets $S_1,\ldots,S_q$ covering all vertices but $q-1$ in each $V_j$, with sizes differing by a…

King's conjecture. There exists a universal constant $\epsilon > 0$ such that every graph contains a stable set hitting every maximal clique of size at least…