Bounding Helly numbers via Betti numbers
LIPIcs
Goaoc, Xavier
Paták, Pavel
Patakova, Zuzana
Tancer, Martin
Wagner, Uli
ddc:510
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 b and d there exists an integer h(b,d) such that the following holds. If F is a finite family of subsets of R^d such that the ith reduced Betti number (with Z_2 coefficients in singular homology) of the intersection of any proper subfamily G of F is at most b for every non-negative integer i less or equal to (d-1)/2, then F has Helly number at most h(b,d). 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 K, some well-behaved chain map from C_*(K) to C_*(R^d). Both techniques are of independent interest.
ACM
2015
info:eu-repo/semantics/conferenceObject
doc-type:conferenceObject
text
https://research-explorer.app.ist.ac.at/record/1512
https://research-explorer.app.ist.ac.at/download/1512/4794
Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. Bounding Helly numbers via Betti numbers. In: Vol 34. ACM; 2015:507-521. doi:<a href="https://doi.org/10.4230/LIPIcs.SOCG.2015.507">10.4230/LIPIcs.SOCG.2015.507</a>
eng
info:eu-repo/semantics/altIdentifier/doi/10.4230/LIPIcs.SOCG.2015.507
'https://creativecommons.org/licenses/by/4.0/'