Reverse mathematics and colorings of hypergraphs

TL;DR

This paper explores the logical strength of hypergraph coloring theorems within subsystems of second order arithmetic, establishing their equivalence to well-known logical systems.

Contribution

It introduces multiple representations for hypergraphs and proves their coloring theorems are equivalent to key subsystems like WKL_0, ACA_0, and Pi^1_1-CA_0.

Findings

Vertex coloring theorems are equivalent to WKL_0, ACA_0, and Pi^1_1-CA_0.

Multiple hypergraph representations are formulated.

Logical strength of hypergraph coloring is characterized.

Abstract

Working in subsystems of second order arithmetic, we formulate several representations for hypergraphs. We then prove the equivalence of various vertex coloring theorems to ${\sf WKL}_0$, ${\sf ACA}_0$ and $\Pi ^1_ 1$-${\sf CA}_0$.