The Open Graph Axiom and Menger's Conjecture

TL;DR

This paper demonstrates that a certain set-theoretic axiom, the perfect set version of the Open Graph Axiom for projective sets, implies Menger's conjecture for projective subsets of real numbers, using only an inaccessible cardinal.

Contribution

It establishes a new connection between the Open Graph Axiom and Menger's conjecture for projective sets, reducing the required consistency strength to an inaccessible cardinal.

Findings

The perfect set version of the Open Graph Axiom implies Menger's conjecture for projective sets.

Consistency strength is reduced to an inaccessible cardinal.

Provides a new set-theoretic approach to Menger's conjecture for projective sets.

Abstract

Menger conjectured that subsets of $\mathbb R$ with the Menger property must be $\sigma$-compact. While this is false when there is no restriction on the subsets of $\mathbb R$, for projective subsets it is known to follow from the Axiom of Projective Determinacy, which has considerable large cardinal consistency strength. We show that the perfect set version of the Open Graph Axiom for projective sets of reals, with consistency strength only an inaccessible cardinal, also implies Menger's conjecture restricted to this family of subsets of $\mathbb R$.