Chromatic number of hypergraph given rank and maximum degree

Question

Lemma 4.3 of the following paper https://faculty.math.illinois.edu/~z-furedi/PUBS/furedi_kahn_poset-dimension.pdf states that a hypergraph with rank a and maximum degree b can be colored with (a-1)b+1 colors but the proof is omitted. How can this be proven? I believe the idea is to induct on a but am unable to complete the proof.

Answer 1

Hint : try to prove it first for usual graphs, it is really the same for hypergraphs.

To color $H$, pick an arbitrary $u \in H$, inductively color $H\setminus u$ with $(a-1)b+1$ colors, then since at most $b$ hyperedges are incident with $u$ in $H$ and each contains at most $a-1$ vertices other than $u$, $u$ has at most $(a-1)b$ neighbors in $H$ so there is at least one color available for $u$ to extend the coloring of $H\setminus u$ into a proper coloring of $H$.