n-Arc Connected Spaces

TL;DR

This paper characterizes n-arc connected spaces, especially graphs, establishing equivalences for 7-ac and analyzing the complexity of characterizing n-ac graphs for smaller n.

Contribution

It provides a comprehensive characterization of n-arc connected graphs and explores the complexity of classifying n-ac graphs for various n.

Findings

7-ac graphs are equivalent to being an injective image of a closed interval.

Characterization of aleph_0-ac continua is provided.

The complexity of classifying n-ac graphs increases for smaller n.

Abstract

A space is `n-arc connected' (n-ac) if any family of no more than n-points are contained in an arc. For graphs the following are equivalent: (i) 7-ac, (ii) n-ac for all n, (iii) continuous injective image of a closed sub-interval of the real line, and (iv) one of a finite family of graphs. General continua that are aleph_0-ac are characterized. The complexity of characterizing n-ac graphs for n=2,3,4,5 is determined to be strictly higher than that of the stated characterization of 7-ac graphs.