4 problems

Let $[k]^n$ be the $k$-ary hypercube. A geometric line is a line in the hypercube in the geometric sense, and the anti-Hales–Jewett number is the minimum number of colors guarantee…

Let $T$ be a tree and let $n$ be an even integer. Let $T\square C_n$ be the Cartesian product of $T$ and the cycle $C_n$, and let $\operatorname{diam}(T\square C_n)$ denote its dia…

Let $G$ and $H$ be graphs, and let $G\square H$ denote their Cartesian product. For a vertex $u$ of a graph, let $\epsilon(u)$ be its eccentricity, let $\operatorname{d}(u,v)$ be t…

Let $G$ be a connected graph with at least $2$ vertices, and let $k$ be a positive integer. Write $\operatorname{aw}(G\square C_n,3)$ for the anti-van der Waerden number of the Car…