6 problems

Let $P_n$ be the path on $n$ vertices, let $P_n\mathbin{\square}P_n$ be the $n\times n$ Cartesian grid, and let $\operatorname{gp}_f$ denote the fractional general position number.…

Let $G$ and $H$ be graphs, let $G\mathbin{\square}H$ denote their Cartesian product, and let $\operatorname{gp}^-(G)$ denote the lower general position number of $G$. Kruft's conje…

Let $G$ be a graph. A set $S\subseteq V(G)$ is a terminal set if $S$ is a general position set of $G$ and adding any vertex $u\in V(G)-S$ to $S$ creates three-in-a-line with $u$ as…

Let $C$ be a $c$-chain in $\mathbb{R}^d$ with $c<d$, and let $U$ be a finite set of points in $\mathbb{R}^d$ in strong general position with respect to $C$. Simplicial-chain realiz…

Let $c,n,d$ be non-negative integers such that $c<d$. Let $C$ be a $c$-chain in $d53d^d$, let $N$ be an $n$-dimensional closed PL manifold, and let $f:N\tod53d^d-\partial C$ be a P…

Near-quadratic growth conjecture. The bound above should hold for every such $m$ and $P$. A proof cannot follow from the paper's main theorem and would require a different construc…