6 problems

Let $d$ be fixed. For even $n$, call a $2d$-dimensional subcube of the hypercube a middle $2d$-cube when its layers range from $n/2-d$ to $n/2+d$. If…

Let $n$ be even, and consider the $4$-dimensional subcubes of the hypercube whose layers range from $n/2-2$ to $n/2+2$; call these the middle 4-cubes. Let…

For fixed integers $s$ and $t$, let $\mathcal D_r(s,t)$ consist of all $r$-sets $A$ containing a fixed $(r-t)$-set $P$ and contained in $P\cup Q$, where $Q$ is a fixed disjoint $s$…

Let $\mathcal A$ be a family of $3$-sets of $[n]$, where $n=7^k$, and suppose that $\mathcal A$ contains no daisy. The power-of-seven daisy bound. … This is proposed as the exact u…

Let $\mathcal D_3$ be the daisy formed from a fixed $1$-set and a disjoint $4$-set, and let $\pi(\mathcal D_3)$ be its limiting Turán density. The 3-daisy density conjecture. … The…

Let $[n]=\{1,\ldots,n\}$. For a set $X$, write $X^{(r)}$ for its $r$-subsets. An $r$-graph is a subset of $X^{(r)}$. Given an $(r-2)$-set $P$ and a disjoint $4$-set $Q$, the $r$-da…