7 problems

Discrete level-set conjecture. There is a result of a similar nature for Lipschitz quotient mappings from $\mathbb{R}^n$ to $\mathbb{R}^n$.

Discreteness conjecture. The mapping $f$ is discrete.

Let $X$ be a metric space and let $Q_o=[0,1]^3$. A Lipschitz map $f\colon Q_o\to X$ is said to factor through a metric tree if there are Lipschitz maps $\psi\colon Q_o\to Z$ and…

Let $Q_0=[0,1]^3$ and let $f\colon Q_0\rightarrow X$ be a $1$-Lipschitz mapping into a metric space. Assume without loss of generality that $X\subseteq\ell_\infty$. A Lipschitz map…

Let $G$ be a connected graph in the class of $n$-vertex graphs, denoted by $\mathcal{G}_n$, and let $P_n$ be the path on $n$ vertices. For a graph $H$, write $\overline{r}_1(H)$ fo…

Let $n,k\geq 1$ and let $J^k(\mathbb R^n)$ be the $k$-th jet space Carnot group. For $n\leq p<n+1$, consider the Sobolev space $W^{1,p}(\mathbb B^{n+1},J^k(\mathbb R^n))$ and the c…

Kalaj's conjecture. The co-Lipschitz conclusion in the second part of Proposition 1.1 remains valid when $\Omega$ is assumed to have only $C^{1,\alpha}$ boundary.