Ramsey degrees and entropy of combinatorial structures

TL;DR

This paper introduces the Ramsey entropy, a new measure of combinatorial complexity for objects in small categories, based on categorical understanding of Ramsey degrees, with properties proven using elementary partitions.

Contribution

It defines and analyzes the Ramsey entropy, connecting structural Ramsey degrees with entropy concepts without requiring advanced categorical machinery.

Findings

Defines the Ramsey entropy as a categorical invariant.

Proves fundamental properties of the Ramsey entropy.

Introduces the Ramsey-Boltzmann entropy as a maximal measure.

Abstract

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.