The linear polychromatic coloring conjecture for hypergraph families

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 hypergraphs of $\mathcal{H}$. Linear polychromatic coloring conjecture. If

$$m_\mathcal{H}(2)<\infty$$

for a hypergraph family $\mathcal{H}$, then

$$m_\mathcal{H}(k)=O(k).$$

The conjecture asserts that finiteness of the two-color threshold forces a linear bound for every number of colors. The surrounding discussion relates this question to shallow hitting sets and the linear bound supplied by the preceding lemma; no resolution is given here.