17 research outputs found
Stabbing boxes with finitely many axis-parallel lines and flats
We give necessary and sufficient condition for an infinite collection of
axis-parallel boxes in to be pierceable by finitely many
axis-parallel -flats, where . We also consider colorful
generalizations of the above result and establish their feasibility. The
problem considered in this paper is an infinite variant of the
Hadwiger-Debrunner -problem.Comment: 13 page
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