6 problems

Extremality conjecture. For all $t,r\geq 2$,

Let $HJ(k,2)$ denote the least dimension such that every coloring of $[k]^{HJ(k,2)}$ with two colors contains a monochromatic combinatorial line. Exact-value conjecture. For all…

Let $[t]^n$ be the set of words of length $n$ over an alphabet of size $t$, let $r$ be the number of colors, and let $\mathrm{HJ}(t,r)$ be the Hales–Jewett number. Let…

Let $[t]^n$ be the set of words of length $n$ over an alphabet of size $t$, let $r$ be the number of colors, and let $\mathrm{HJ}(t,r)$ be the Hales–Jewett number. The diagonal is…

Let $[t]^n$ be the set of words of length $n$ over an alphabet of size $t$, let $r$ be the number of colors, and let $\mathrm{HJ}^{[k]}(t,r)$ denote the least dimension forcing a m…

Collapse conjecture.