Goaoc, Xavier; Paták, Pavel; Patakova, Zuzana ; Tancer, Martin ; Wagner, UliIST Austria
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.
PP, ZP and MT were partially supported by the Charles University Grant GAUK 421511. ZP was partially supported by the Charles University Grant SVV-2014-260103. ZP and MT were partially supported by the ERC Advanced Grant No. 267165 and by the project CE-ITI (GACR P202/12/G061) of the Czech Science Foundation. UW was partially supported by the Swiss National Science Foundation (grants SNSF-200020-138230 and SNSF-PP00P2-138948). Part of this work was done when XG was affiliated with INRIA Nancy Grand-Est and when MT was affiliated with Institutionen för matematik, Kungliga Tekniska Högskolan, then IST Austria.
507 - 521
SoCG: Symposium on Computational Geometry
2015-06-22 – 2015-06-25
Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. Bounding Helly numbers via Betti numbers. In: Vol 34. ACM; 2015:507-521. doi:10.4230/LIPIcs.SOCG.2015.507
Goaoc, X., Paták, P., Patakova, Z., Tancer, M., & Wagner, U. (2015). Bounding Helly numbers via Betti numbers (Vol. 34, pp. 507–521). Presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands: ACM. https://doi.org/10.4230/LIPIcs.SOCG.2015.507
Goaoc, Xavier, Pavel Paták, Zuzana Patakova, Martin Tancer, and Uli Wagner. “Bounding Helly Numbers via Betti Numbers,” 34:507–21. ACM, 2015. https://doi.org/10.4230/LIPIcs.SOCG.2015.507.
X. Goaoc, P. Paták, Z. Patakova, M. Tancer, and U. Wagner, “Bounding Helly numbers via Betti numbers,” presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands, 2015, vol. 34, pp. 507–521.
Goaoc X, Paták P, Patakova Z, Tancer M, Wagner U. 2015. Bounding Helly numbers via Betti numbers. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34, 507–521.
Goaoc, Xavier, et al. Bounding Helly Numbers via Betti Numbers. Vol. 34, ACM, 2015, pp. 507–21, doi:10.4230/LIPIcs.SOCG.2015.507.
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