Ramsey degrees and entropy of combinatorial structures

Dragan Mašulović

Published 2022-03-31Version 1

In this paper we propose a way to compute entropy of a finite structure not as a measure of statistical, but as a measure of its combinatorial complexity. Close connections between various notions of entropy and the apparatus of category theory have been observed already in the 1980's and more vigorously developed in the past ten years. In this paper we add to the body of arguments in favor of the categorical treatment of entropy. Our entropy function, the Ramsey entropy, is a real 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. Along the way we prove several properties of Ramsey degrees that suggest that Ramsey degrees actually behave as a measure of diversity for combinatorial objects. We conclude the paper with the discussion of the maximal Ramsey entropy on a category that we refer to as the Ramsey-Boltzmann entropy.