11 problems

Let $\mathcal{H}$ be a hereditary family, with $\mathcal{H}^{(r)}$ and $\mathcal{H}^{(s)}$ its $r$th and $s$th levels. For cross-intersecting subfamilies, let…

Cross-intersection conjecture for hereditary families. If $\mu(\mathcal{H})\geq 2r$ and $\mathcal{A},\mathcal{B}\subseteq\mathcal{H}^{(r)}$ are cross-intersecting, then

Let $\mathcal C$ be a hereditary family of sets whose maximal sets all have size at least $n$, and let $\mathcal C^{(k)}$ denote its family of $k$-element sets. Let…

Polychromatic finiteness conjecture. If $m_2(\mathcal F)<\infty$, then $m_k(\mathcal F)<\infty$ for every $k$ for any hereditary family $\mathcal F$.

Let $\mathcal{H}$ be a hereditary family, let $\mathcal{H}^{(r)}$ and $\mathcal{H}^{(s)}$ be its levels, and suppose $\mathcal{A}\subseteq\mathcal{H}^{(r)}$ and…

For a hereditary family $\mathcal{H}$, write $\mathcal{H}^{(r)}$ for its $r$th level and define $\eta(r,s,t)$ to be the least integer $n$ such that every hereditary family with…

Let $\mathcal{H}$ be a hereditary family, and let $\mathcal{H}^{(r)}(I)=\{A\in\mathcal{H}^{(r)}:I\subseteq A\}$ denote the $r$-star centered at $I$. Let…

Sharp-threshold conjecture. The maximum-product conjecture holds with

Maximum-product conjecture. There exists an integer $n^*_{\rm P}(r,t)$ such that, whenever $\mu(\mathcal{H})\geq n^*_{\rm P}(r,t)$, there is some $T\in\mathcal{H}^{(t)}$ for which

Equality conjecture for cross-intersecting families. If $\mu(\mathcal{H})>2r$, then equality in

Let $1\leq t\leq r$, let $\emptyset\neq S\subseteq [r]$, and let $\mathcal{H}$ be a hereditary family with covering number $\mu(\mathcal{H})\geq (t+1)(r-t+1)$. Write…