130 research outputs found
Bounding Helly numbers via Betti numbers
We show that very weak topological assumptions are enough to ensure the
existence of a Helly-type theorem. More precisely, we show that for any
non-negative integers and there exists an integer such that
the following holds. If is a finite family of subsets of such that for any
and every
then has Helly number at most . Here
denotes the reduced -Betti numbers (with singular homology). These
topological conditions are sharp: not controlling any of these first Betti numbers allow for families with unbounded Helly number.
Our proofs combine homological non-embeddability results with a Ramsey-based
approach to build, given an arbitrary simplicial complex , some well-behaved
chain map .Comment: 29 pages, 8 figure
On Minc's sheltered middle path
This paper shows that a construction, which was introduced by Piotr Minc in
connection with a problem that came from Helly type theorems and that allows to
replace three PL-arcs with a "sheltered middle path", can in the case of
general (non-PL) paths result in the topologist's sine curve.Comment: 18 pages, 3 figure
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