  This paper studies the online vector bin packing (OVBP) problem and the
related problem of online hypergraph coloring (OHC). Firstly, we use a double
counting argument to prove an upper bound of the competitive ratio of
$FirstFit$ for OVBP. Our proof is conceptually simple, and strengthens the
result in Azar et. al. by removing the dependency on the bin size parameter.
Secondly, we introduce a notion of an online incidence matrix that is defined
for every instance of OHC. Using this notion, we provide a reduction from OHC
to OVBP, which allows us to carry known lower bounds of the competitive ratio
of algorithms for OHC to OVBP. Our approach significantly simplifies the
previous argument from Azar et. al. that relied on using intricate graph
structures. In addition, we slightly improve their lower bounds. Lastly, we
establish a tight bound of the competitive ratio of algorithms for OHC, where
input is restricted to be a hypertree, thus resolving a conjecture in
Nagy-Gyorgy et. al. The crux of this proof lies in solving a certain
combinatorial partition problem about multi-family of subsets, which might be
of independent interest.
