9 problems

Collapse conjecture.

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 $HJ^s(k,2)$ denote the least dimension such that every two-coloring in the line-symmetric Hales–Jewett variant on an alphabet of size $k$ contains a monochromatic line. Exact-v…

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 $\mathrm{HJ}^{[k]}(t,r)$ denote the least dimension forcing a m…

For positive integers $n,d,k$ with $k\geqslant 2$, let $W(n,d,k)$ be the set of maps $w\colon [n]^d\to [k]$, whose elements are called polynomial words over $[k]$. A polynomial var…

Let $n$ be a positive integer, and write $\mathbb{F}_2^{\binom{[n]}{\leqslant 2}}$ for the vector space of functions on subsets of $[n]$ of size at most $2$. Let…

Let $[3]^n$ be the set of words of length $n$ over a three-element alphabet, and let a combinatorial line be a collection of words obtained by assigning the same symbol to a nonemp…