7 problems

Let $\mathcal F$ be a hereditary family of hypergraphs, and let $m_k(\mathcal F)$ denote the least edge-size threshold guaranteeing a polychromatic $k$-coloring of every member of…

Let $\mathcal F$ be the family of bottomless rectangles in the plane, and let $m_k^*(\mathcal F)$ denote the corresponding threshold for decomposing coverings into $k$ coverings. B…

Let $\mathcal F$ be a hereditary family of hypergraphs, and let $m_k(\mathcal F)$ be the least threshold guaranteeing a polychromatic $k$-coloring for every $m$-heavy member of…

Let $\mathcal F$ be a hereditary family of hypergraphs. For $k\ge1$, let $m_k(\mathcal F)$ be the smallest integer $m$ such that every $m$-heavy hypergraph in $\mathcal F$ has a po…

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 hypergraph family, and let $m_\mathcal{H}(k)$ denote the minimum number of vertices needed to guarantee a polychromatic $k$-coloring in the relevant hypergra…

Let ${\mathcal T}$ be the family of all translates of a fixed compact convex set in the plane, and let $\widetilde{f}_{\mathcal T}(2)$ be the minimum number such that every finite…