8 problems

Does there exist a biplanar graph $G=(V,E)$ such that $|V|=19$ and $\alpha(G)=2$? Here, biplanar means that $E$ can be partitioned into the edge sets of two planar graphs on the co…

For every integer $n\ge 1$ and every convex body $K\subset\mathbb{R}^n$, let $E$ be a maximum-volume ellipsoid contained in $K$, let $c$ be the center of $E$, and define…

For every integer $n\ge 2$, let $X=(X_1,\ldots,X_n)$ be a centered Gaussian vector satisfying $\operatorname{Var}(X_i)=1$ for every $i$. Let $Z_1,\ldots,Z_n$ be independent standar…

For every finite simple connected graph $G$ with $|V(G)|>1$, if $\alpha(G)\le 1+\ell_{\mathrm{avg}}(G)$, where $\alpha(G)$ is the independence number of $G$ and…

For every finite simple connected graph $G$ containing a cycle, define $t(G)=\max\{|S|:S\subseteq V(G)\text{ and }G[S]\text{ is a tree}\}$, let $g(G)$ be the length of a shortest c…

For every finite simple connected graph $G=(V,E)$, let $N_G(v)$ denote the neighborhood of $v$, let $\alpha(G[N_G(v)])$ be the independence number of the subgraph induced by…

For $N=2^n$, let $s\in\mathbb{Z}_N$ be hidden. Given polynomially many independent copies of the mixed coset state…

Conjecture 9.1 proposes the identity … where $\mathsf{LD}(0,n)^{\ast k,\circ r}$ is the set of positively labelled Dyck paths of size $n$ with $k$ decorated rises and $r$ decorated…