21 problems

Lindner's conjecture. Any partial Steiner triple system of order $m$ can be embedded in a Steiner triple system of order $n$ whenever

Let $K_m^h$ and $K_n^h$ be complete $h$-uniform hypergraphs on $m$ and $n$ vertices, respectively, where $n>m$. A 1-factorization is an edge-coloring whose color classes are $1$-re…

Let $k$ be a Hilbertian field. A finite split embedding problem over $k$ is an epimorphism … where $G$ is finite, $L/k$ is Galois, and there is an embedding…

Let $Delta_n^k$ be the union of the $k$-dimensional faces of the $n$-simplex, let $M$ be a closed $(k-1)$-connected PL $2k$-manifold, and let $chi(M)$ denote its Euler characterist…

Abhyankar–Sathaye embedding conjecture. If $H$ is isomorphic to a coordinate hypersurface, then $H$ is equivalent to it under an automorphism of the ambient affine space.

A graph group $G(\Gamma)$ is unconcealable if, whenever it embeds into a graph group $G(\Omega)$, the graph $\Gamma$ embeds into $\Omega$. The free group of rank $2$ and all free a…

Let $N$ be an oriented $\mathbb{S}^1$-bundle over a closed, oriented $3$-manifold $M$. The bundle has an Euler class, and suppose this Euler class is even. The main result gives a…

Let $G\subset S_n$ be a transitive permutation group and let $k$ be a number field. Set … Consider the finitely many continuous surjective pairs $(\pi,\phi)$ satisfying the cycloto…

Let $G\subset S_n$ be a finite group and let $Q$ be a global field with $(|G|,\operatorname{char}(Q))=1$. Set … For a cyclotomic subfield $F\subset Q(\mu_d)$, a surjection…

Generalized Menger embedding conjecture. The van Kampen obstruction of $K\times L$ is non-zero.

Cristofaro-Gardiner–Frenkel–Schlenk conjecture. For $a$ in the domain of the linear steps, one has

Let $\mathcal{F}$ be a partial $(n,k)$-design, meaning a collection of $k$-element blocks on an $n$-element set in which every pair is covered at most once, and let an embedding be…

Let $X$ be a separable Banach space whose conjugate space $X^*$ is non-separable. The $ell^1$-embedding conjecture. The space $X$ should contain an isomorphic copy of $\ell^1$. Thi…

Marker-free embedding conjecture. If, for every subgroup $A\subset \mathbb{Z}^k$, the mean dimension of $X_A$ with respect to the $(\mathbb{Z}^k/A)$-action is smaller than $D/2$, t…

Let $(X,\mathbb{Z}^k)$ be a dynamical system, and for each subgroup $A$ of $\mathbb{Z}^k$ define … The quotient group $\mathbb{Z}^k/A$ acts naturally on $X_A$; when the quotient is…

Codimension-1 Grassmannian embedding conjecture. The only codimension-$1$ smooth embedding

Let $(B_t\colon t\in\mathbb{R})$ be a two-sided Brownian motion, and let $T$ be an unbiased shift embedding a probability measure orthogonal to the initial distribution. The shift…

A split finite embedding problem (FSEP) over a field $K$ consists of a finite Galois extension and a split surjection of finite groups associated with it; it is solvable if there i…

Let $K$ be a Hilbertian field. A finite split embedding problem over a field $E$ consists of a finite Galois extension $F_1/E$ and a split epimorphism of finite groups … A solution…

Let $K$ be a Hilbertian field, meaning that every irreducible polynomial $f(T,X)$ that is separable in $X$ admits a specialization $a\in K$ such that $f(a,X)$ remains irreducible.…

Let $k$ and $m$ be positive integers. A simple partial $k$-latin square is a partial $k$-latin square in which no symbol is repeated within a cell. Eventual embedding conjecture. T…