  The description complexity of a model is the length of the shortest formula
that defines the model. We study the description complexity of unary structures
in first-order logic FO, also drawing links to semantic complexity in the form
of entropy. The class of unary structures provides a simple way to represent
tabular Boolean data sets as relational structures. We define structures with
FO-formulas that are strictly linear in the size of the model as opposed to
using the naive quadratic ones, and we use arguments based on formula size
games to obtain related lower bounds for description complexity. We also obtain
a precise asymptotic result on the expected description complexity of a
randomly selected structure. We then give bounds on the relationship between
Shannon entropy and description complexity. We extend this relationship also to
Boltzmann entropy by establishing an asymptotic match between the two
entropies. Despite the simplicity of unary structures, our arguments require
the use of formula size games, Stirling's approximation and Chernoff bounds.
