5 problems

Let $\mathcal{F}(A)$ denote the family of finite subsets of $A$, and let $[N]=\{1,\dots,N\}$. For $k,d\in\mathbb{N}$ with $k\geqslant2$ and every $\delta>0$, consider sets…

Let $s,d_1,\dots,d_s$ be positive integers, and let $\delta>0$. For a positive integer $n$, let … be the power set, with density measured relative to its cardinality. The case…

Let $k,s,d_1,\dots,d_s$ be positive integers and let $\delta>0$. For a positive integer $n$, consider the product set … A subset has density at least $\delta$ if its cardinality is…

Leader–Russell–Walters Hales–Jewett conjecture. There exist positive integers $d$ and $N$ such that, for every $r$-coloring of $G^N$, there exist a nonempty $H\subseteq G$ and an…

Let $\mathcal{I}(m,r)$ denote the interval parameter studied in the paper, and let $H\!J(m-1,r)$ denote the Hales–Jewett number for alphabet size $m-1$ and number of colours $r$. C…