I can. But my proof uses a theorem (which I do not reveal yet to avoid influencing you) and it feels like an overkill, so I wonder if there is a simple proof. Now the problem.
Suppose we have a hypergraph on n vertices with n-1 edges. Can we color some (at least one) of its vertices with red and blue such that every hyperedge either contains both colors, or all of its vertices are uncolored?
The motivation is this question. In fact, one can similarly define partial k-polychromatic coloring, where every hypergraph with less than n/(k-1) edges seems to be partially k-colorable, if my proof is correct. Note also that these bounds are tight as shown by disjoint edges of size k-1. And this cannot be improved by requiring some lower bound on the size of each hyperedge, as we observed yesterday with my usual set of friends, Cory, Dani, Keszegh, Nathan and Patkos.
Please do not post links to Beck-Fiala and other well-known, similar theorems! At the moment I am only interested in a short, elementary proof of my question.