par Cardinal, Jean 
;Tóth, Csaba C.D.;Wood, David
Référence Journal of geometry, 108, 1, page (33-43)
Publication Publié, 2017-04
;Tóth, Csaba C.D.;Wood, David Référence Journal of geometry, 108, 1, page (33-43)
Publication Publié, 2017-04
Article révisé par les pairs
| Résumé : | Erdős asked what is the maximum number α(n) such that every set of n points in the plane with no four on a line contains α(n) points in general position. We consider variants of this question for d-dimensional point sets and generalize previously known bounds. In particular, we prove the following two results for fixed d:Every set H of n hyperplanes in Rd contains a subset S⊆ H of size at least c(nlog n) 1 / d, for some constant c= c(d) > 0 , such that no cell of the arrangement of H is bounded by hyperplanes of S only.Every set of cqdlog q points in Rd, for some constant c= c(d) > 0 , contains a subset of q cohyperplanar points or q points in general position.Two-dimensional versions of the above results were respectively proved by Ackerman et al. [Electronic J. Combinatorics, 2014] and by Payne and Wood [SIAM J. Discrete Math., 2013]. |
