Steenrod Lengths and a Problem of Vakil

TL;DR

This paper provides a combinatorial formula for the Steenrod length function of real projective spaces, connecting algebraic topology with graph theory, and resolves a question posed by Vakil.

Contribution

It introduces a combinatorial interpretation of the Steenrod length function using directed graphs and explicitly computes it in terms of binary classes and Vakil numbers.

Findings

Derived explicit formulas for the function f(n)

Connected Steenrod algebra actions with directed graph paths

Resolved Vakil's question on Steenrod length characterization

Abstract

We give an explicit combinatorial description of the function $f(n)$ governing the Steenrod length of real projective spaces $\mathbb{RP}^n$. This function arises in stable homotopy theory through the action of Steenrod squares on mod-$2$ cohomology and is closely related to the ghost length, which measures the minimal number of spheres required to construct a space up to homotopy. Building on the directed graphs $T_n$ introduced by Vakil to encode degree constraints for Steenrod operations, we interpret $f(n)$ as the length of the longest directed path starting at $n$. Using this framework, we resolve a question posed by Vakil by deriving concrete combinatorial formulas for $f(n)$ in terms of binary classes and a distinguished family of integers, which we call Vakil numbers.