9 problems

Let $V$ be an $n$-dimensional vector space, and let ${\rm CB}(V)$ be the simplicial complex whose simplices are collections of proper non-zero subspaces of $V$ with a common basis.…

Let $V$ be a finite-dimensional vector space over a field $k$. Let ${\mathcal{PD}}(V)$ be the poset of non-empty partial decompositions of $V$, ordered by refinement, and let…

For each positive integer $n$, consider the vector space $\mathbb{F}_2^n$ and a two-coloring of its nonzero vectors. A subspace $H\leq\mathbb{F}_2^n$ is monochromatic when all its…

For each integer $s\geqslant2$, let $R_{\mathbb F_{2}}(s,d)$ denote the corresponding vector-space Ramsey number over $\mathbb F_{2}$, with parameter $s$ and target dimension $d$.…

Let $R_{\mathbb F_{2}}(2,d)$ denote the least $n$ such that every red-blue coloring of $\mathbb F_{2}^{n}$ contains a monochromatic affine subspace of the relevant prescribed type…

Let $\mathcal{A}$ be an $\omega$-categorical structure, let $X$ range over definable sets, and let $\mathbf{K}^X$ denote the associated dual vector space. Let $\overline{X}$ denote…

Algebraic-closure and uniform-finiteness conjecture. In $V^\pi$, the algebraic closure of a set $A\subset V$ is

Let $V_n(q)$ be an $n$-dimensional vector space over the finite field with $q$ elements, and let ${\bf P}$ be a non-trivial partition of $V_n(q)$. Suppose that $t$ is the minimum d…

Let $M$ and $N$ be finite-dimensional $\mathbb{R}$-computable $\mathbb{R}$-vector spaces with the same dimension. Computable categoricity conjecture. There should be a $\mathbb{R}$…