A note on the Erdős-Selfridge potential

The following small result has been sitting untouched in my research folder since December, 2014. As I have not been able to produce other interesting results in the field it has remained unpublished until now. A hypergraph is properly 2-colorable if its vertex set can be colored blue and red such that there is no monochromatic edge. Lemma 1 (TRM, 2014). Let $\mathcal{H}$ be an $r$-uniform hypergraph on $n$ vertices.