16 problems

Let $r\in(0,1]$. A polytope $P=\operatorname{conv}(V)\subset\mathbb{R}^n$ contains $rB_2^n$ when $rB_2^n\subset P$, and $N(P,B_2^n)$ denotes the covering number of $P$ by translate…

Let $n\geq 5$ and let $\lambda=(n-k,k)$, where $1\leq k\leq \lfloor\frac n2\rfloor$. Write $\chi_\lambda$ for the irreducible character of $S_n$ indexed by $\lambda$, let…

Let $V$ be a $(K,n)$ regular set in $\mathcal{B}^N$, and let $\mathcal{N}(V,\eps)$ denote its covering number at scale $\eps$. Covering-number dependence conjecture. There exists a…

Pinnable rainbow matching conjecture. If

General pinnable covering conjecture. If $A\cup B$ is pinnable, then

Pinnable covering conjecture. If $A\cup B$ is pinnable, then

Gyárfás–Lehel conjecture. Then

Let $G$ be an $n$-vertex graph, let $T(G)$ denote its number of triangles, and let $\tau_{\triangle}(G)$ be the minimum size of a vertex set meeting every triangle. For integers…

Refined intersecting-segment conjecture. If $H$ is an intersecting $r$-segment hypergraph and $r\geq 5$, then

Let $\mathscr{E}$ be the set of all integers that are covering numbers, meaning integers of the form $\sigma(G)$ for some group $G$. Non-covering-number infinitude conjecture. The…

A finite group $G$ is $c3$-elementary if $c3(G)$ is finite and $c3(H)<c3(G)$ for every proper quotient $H$ of $G$. A group is monolithic primitive if it is primitive and has a uniq…

Let $K\subseteq\mathbb{R}^{n}$ be a convex body with nonempty interior, and let $N(K,L)$ denote the least number of translates of $L$ needed to cover $K$. A parallelotope is an aff…

Covering-number duality conjecture. There are universal constants $c,C>0$ such that, for every dimension $n$ and every pair of symmetric convex bodies $K,L\subset\mathbb R^n$,

Let $H$ be an $r$-partite hypergraph with matching number $u(H)=k$. A set is contained in a side or in an edge if it is a subset of one of the sides or of one edge, respective…

Let $H$ be an intersecting $r$-partite hypergraph with sides $V_1,\ldots,V_r$. A cover is a set of vertices meeting every edge. The structured-cover conjecture. There exists either…

Let $K\subseteq\mathbb{R}^{n}$ be a convex body, and let $N_{\omega}(K,T)$ denote its weighted covering number. Weighted Levi–Hadwiger covering conjecture. … Moreover, equality hol…