  Close connections between various notions of entropy and the apparatus of
category theory have been observed already in the 1980s and more vigorously
developed in the past ten years. The starting point of the paper is the recent
categorical understanding of structural Ramsey degrees, which then leads to a
way to compute entropy of an object in a small category not as a measure of
statistical, but as a measure of its combinatorial complexity. The new entropy
function we propose, the Ramsey entropy, is a real-valued invariant of an
object in an arbitrary small category. We require no additional categorical
machinery to introduce and prove the properties of this entropy. Motivated by
combinatorial phenomena (structural Ramsey degrees) we build the necessary
infrastructure and prove the fundamental properties using only special
partitions imposed on homsets. We conclude the paper with the discussion of the
maximal Ramsey entropy on a category that we refer to as the Ramsey-Boltzmann
entropy.
