An introduction to higher walks

TL;DR

This paper introduces higher-dimensional extensions of Todorcevic's method of walks on ordinals, exploring their properties, combinatorial significance, and potential for advancing higher-dimensional infinitary combinatorics.

Contribution

It develops a framework for higher walks, including new functions like Tr_2, and investigates their properties and implications for ordinal combinatorics and cohomology.

Findings

Higher-dimensional functions Tr_n and ρ_2^n satisfy key desiderata.

ρ_2^n determines n-dimensional orderings and cohomology elements.

Tr_2 exhibits significant combinatorial richness and novelty.

Abstract

The following is an introduction to the study of higher walks, by which we mean a family of higher-dimensional extensions of Todorcevic's method of walks on the ordinals. After a brief review of this method, including, for example, definitions of the classical functions $\mathrm{Tr}$ and $\rho_2$ induced by a choice of $C$-sequence, we record a shortlist of desiderata for such extensions, along with $(n+1)$-dimensional functions $\mathrm{Tr}_n$ and $\rho_2^n$ (induced by a choice of higher-dimensional $C$-sequence) which we show to satisfy the bulk of them. Much of the interest of these higher walks functions lies in their affinity, as in the classical $n=1$ case, for the ordinals $\omega_n$ (we show, for example, that $\rho^n_2$ determines both $n$-dimensional linear orderings and $n$-coherent families on $\omega_n$, and that higher walks define nontrivial elements of the $n^{\mathrm{th}}$ cohomology groups of $\omega_n$), and in the questions that they thereby raise both about the combinatorics of the latter and about higher-dimensional infinitary combinatorics more generally; we collect the most prominent of these questions in our conclusion. These objects are also, though, of a sufficient combinatorial richness to be of interest in their own right, as we have underscored via an extended study of the first genuine novelty among them, the function $\mathrm{Tr}_2$.