6 problems

Let $\delta>0$. Consider the collection of non-oriented graphs on the vertex set $[n]$, with density measured relative to the number of such graphs. Gowers' clique-difference conje…

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…

Fix $k\in\mathbb{N}$ and $0<delta<1$. Let $Emb(1,delta,k)$ be the smallest $N$ such that every subset $\mathcal{D}\subset\{0,1\}^N$ of density at least $delta$ contains the image o…

Polynomial difference conjecture. For $n$ large enough depending on $s,d_1,\dots,d_s,\delta$ only, every subset of this power set with density at least $\delta$ contains distinct s…

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…