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…

Let $0<\delta<1$. For a positive integer $n$, identify elements of $\{0,1\}^{n^2}$ with fillings of an $n\times n$ grid. A combinatorial line in this space consists of two elements…