4 problems

Restricted density polynomial Hales–Jewett conjecture. There exists $N=N(d,\delta)\in\mathbb N$ such that, for every $S\subseteq\mathcal P(\{1,\ldots,N\}^d)$ satisfying

Let $d$ be a positive integer and let $\delta>0$. For a finite set, density means cardinality divided by the cardinality of the ambient set. The notation $[n]^d$ denotes the $d$-fo…

Let $k,d$ be positive integers, let $\delta>0$, and let $n$ be a positive integer. Write $[n]=\{1,\ldots,n\}$ and … For a subset of a finite set, its density is its cardinality divi…

For positive integers $k$ and a positive real number $0<\delta<1$, define … For $a\in K[n]$, let $\gamma\subset[N]$ and interpret $\gamma^r$ as the corresponding $r$-fold Cartesian…