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 subinterval of the real line, and (iv) one of a finite family of graphs. General continua that are ℵ₀-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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-2-5,
author = {Benjamin Espinoza and Paul Gartside and Ana Mamatelashvili},
title = {n-Arc connected spaces},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {221-240},
zbl = {1271.54065},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-2-5}
}
Benjamin Espinoza; Paul Gartside; Ana Mamatelashvili. n-Arc connected spaces. Colloquium Mathematicae, Tome 131 (2013) pp. 221-240. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm130-2-5/