Abstract
Erdős asked what is the maximum number \({\alpha(n)}\) such that every set of \({n}\) points in the plane with no four on a line contains \({\alpha(n)}\) 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 \({\mathcal{H}}\) of \({n}\) hyperplanes in \({\mathbb{R}^d}\) contains a subset \({S\subseteq \mathcal{H}}\) of size at least \({c \left(n \log n\right)^{1/d}}\), for some constant \({c=c(d)> 0}\), such that no cell of the arrangement of \({\mathcal{H}}\) is bounded by hyperplanes of \({S}\) only.

Every set of \({cq^d\log q}\) points in \({\mathbb{R}^d}\), for some constant \({c=c(d)> 0}\), contains a subset of \({q}\) cohyperplanar points or \({q}\) points in general position.
Twodimensional 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].
Similar content being viewed by others
References
Ackerman, E., Pach, J., Pinchasi, R., Radoičić, R., Tóth, G.: A note on coloring line arrangements. Electron. J. Comb. 21(2), #P2.23 (2014)
Bose P., Cardinal J., Collette S., Hurtado F., Korman M., Langerman S., Taslakian P.: Coloring and guarding arrangements. Discret. Math. Theor. Comput. Sci. 15(3), 139–154 (2013)
Cardinal, J., Felsner, S.: Covering partial cubes with zones. In: Proceedings of the 16th Japan conference on discrete and computational geometry and graphs (JCDCG\({^2}\) 2013), Lecture notes in computer science. Springer, Berlin (2014)
Elekes, G., Tóth, C.D.: Incidences of nottoodegenerate hyperplanes. In: Proceedings of the ACM symposium on computational geometry (SoCG), pp. 16–21 (2005)
Erdős P.: On some metric and combinatorial geometric problems. Discret. Math. 60, 147–153 (1986)
Füredi Z.: Maximal independent subsets in Steiner systems and in planar sets. SIAM J. Discret. Math. 4(2), 196–199 (1991)
Furstenberg H., Katznelson Y.: A density version of the Hales–Jewett theorem for \({k=3}\). Discret. Math. 75(13), 227–241 (1989)
Furstenberg H., Katznelson Y.: A density version of the Hales–Jewett theorem. J. d’analyse Math. 57, 64–119 (1991)
Gowers, T.: A geometric Ramsey problem. http://mathoverflow.net/questions/50928/ageometricramseyproblem (2012)
Halperin D.: Arrangements. In: Goodman, J.E., O’Rourke, J. (eds) Handbook of Discrete and Computational Geometry, chapter 24., 2nd edn., CRC Press, Boca Raton (2004)
Kostochka A.V., Mubayi D., Verstraëte J.: On independent sets in hypergraphs. Random Struct. Algorithms 44(2), 224–239 (2014)
Milićević, L.: Sets in almost general position (2015). arXiv:1601.07206
Payne M.S., Wood D.R.: On the general position subset selection problem. SIAM J. Discret. Math. 27(4), 1727–1733 (2013)
Polymath D.H.J.: A new proof of the density Hales–Jewett theorem. Ann. Math. 175(2), 1283–1327 (2012)
Solymosi J., Stojaković M.: Many collinear \({k}\)tuples with no \({k+1}\) collinear points. Discret. Comput. Geom. 50, 811–820 (2013)
Spencer J.: Turán’s theorem for \({k}\)graphs. Discret. Math. 2(2), 183–186 (1972)
Szemerédi E., Trotter W.T.: Extremal problems in discrete geometry. Combinatorica 3(3), 381–392 (1983)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research of Wood is supported by the Australian Research Council.
Rights and permissions
About this article
Cite this article
Cardinal, J., Tóth, C.D. & Wood, D.R. General position subsets and independent hyperplanes in dspace. J. Geom. 108, 33–43 (2017). https://doi.org/10.1007/s0002201603235
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s0002201603235