TITLE: Can you prove that hypergraphs with n-1 edges are partially 2 colorable?
QUESTION [6 upvotes]: 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.

REPLY [4 votes]: The proof of the statement of the question is implicitly contained, along the lines of
the proof by Ilya Bogdanov, in Paul Seymour's MSc thesis, where the relevant part is
published in

Paul D. Seymour, On the two-colouring of hypergraphs, The Quarterly Journal of Mathematics (Oxford University Press), 1974, Volume 25, Pages 303-312.

If the hypergraph 2-colouring problem is translated (in the canonical way) into a satisfiability problem (SAT problem), then the statement is precisely that there
exists a so-called "autarky". The theory of autarkies is outlined in my Handbook
of Satisfiability article on autarkies, available at
http://www.cs.swan.ac.uk/~csoliver/papers.html#Handbook2009MUAUT
(see Subsections 11.4.5 and 11.12 for connections to the problem at hand).
An article extending Seymour's result is
http://cs.swan.ac.uk/~csoliver/papers.html#SNSSAT2007