On a topological fractional Helly theorem [article]

Stephan Hell
2005 arXiv   pre-print
We prove a new fractional Helly theorem for families of sets obeying topological conditions. More precisely, we show that the nerve of a finite family of open sets (and of subcomplexes of cell complexes) in R^d is k-Leray where k depends on the dimension d and the homological intersection complexity of the family. This implies fractional Helly number k+1 for families F. Moreover, we obtain a topological (p,q)-theorem. Our result contains the (p,q)-theorem for good covers of Alon, Kalai,
more » ... , and Meshulam (2003) as a special case. The proof uses a spectral sequence argument. The same method is then used to reprove a homological version of a nerve theorem of Bjoerner.
arXiv:math/0506399v1 fatcat:zedt4cghpbcphnnasoc26mbsvy