7 problems

Keszegh–Pálvölgyi linear bound conjecture. For every $k$ and every hereditary family $\mathcal F$,

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…

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…

Let $\mathcal{H}$ be a hereditary family, let $\mu(\mathcal{H})$ denote the size of a smallest base of $\mathcal{H}$, and let $\mathcal{H}^{(r)}$ be its $r$th level. The $r$th leve…

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 $1\leq t\leq r$, 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…