6 problems

A partial Meshalkin sequence of length $p$ in the projective geometry $\mathbb{P}^{\,n-1}(q)$ is a sequence of flats $a=(a_1,\dots,a_p)$ such that…

Linear-dimension strengthening. The multicoloured geometric Erdős–Hajnal conjecture and the geometric Erdős–Hajnal conjecture for induced restrictions should both hold with…

Geometric Erdős–Hajnal conjecture. For every prime power $q$ and all $k\in\mathbb N$ and $H\subseteq\operatorname{PG}(k-1,q)$, there exist $\delta,C>0$ such that, for all…

Multicoloured geometric Erdős–Hajnal conjecture. For every prime power $q$, all $k,s_0,s\in\mathbb N$ with $s_0\le s$, and every $s_0$-colouring $c_0$ of $\operatorname{PG}(k-1,q)$…

Projective-geometry embedding conjecture. Every clean tangled clutter embeds a projective geometry over the two-element field.

For a simple binary matroid $N$, its critical threshold is defined using the least density above which the critical number of $N$-free simple binary matroids is bounded. Let…