Finite combinatorics and computability theory

TL;DR

This paper links finite combinatorial objects to algorithmic reductions and uses computability theory to derive new results in finite combinatorics.

Contribution

It introduces a novel approach connecting combinatorial existence problems with computability and reduction techniques.

Findings

Finite combinatorial objects relate to algorithmic problem reductions.

Computability theory refines understanding of combinatorial existence.

New results in finite combinatorics derived from computability techniques.

Abstract

We prove that the existence of finite combinatorial objects such as affine planes, mutually orthogonal Latin squares, and resolvable balanced incomplete block designs can be reformulated as the existence of certain algorithmic reductions between problems related to the pigeonhole principle. We then study the latter using counting arguments and computability theory. In particular, we demonstrate that computability theoretic techniques can be used to refine and prove new results in finite combinatorics.