Abstract
We establish the following result related to Erdős’s problem on distinct distances. Let V be an n-element planar point set such that any p members of V determine at least \(\left( {\begin{array}{*{20}{c}} p \\ 2 \end{array}} \right) - p + 6\) distinct distances. Then V determines at least \(n^{\tfrac{8} {7} - o(1)}\) distinct distances, as n tends to infinity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. A. Behrend: On sets of integers which contain no three terms in arithmetical progression, Proc. Nat. Acad. Sci. U. S. A. 32 (1946), 331–332.
P. Brass, W. Moser and J. Pach: Research Problems in Discrete Geometry, Berlin, Germany, Springer-Verlag, 2005.
A. Dumitrescu: On distinct distances among points in general position and other related problems, Period. Math. Hungar. 57 (2008), 165–176.
P. Erdős: On sets of distances of n points, Amer. Math. Monthly 53 (1946), 248–150.
P. Erdős: On some metric and combinatorial geometric problems, Discrete Math. 60 (1986), 147–153.
P. Erdős: Some of my recent problems in combinatorial number theory, geometry and combinatorics, in: Graph Theory, Combinatorics, Algorithms and Applications, vol. 1 (Y. Alavi et al., eds.), Wiley 1995, 335–349.
P. Erdős, Z. Füredi, J. Pach and I. Z. Ruzsa: The grid revisited, Discrete Math. 111 (1993), 189–196.
P. Erdős and A. Gyárfás: A variant of the classical Ramsey problem, Combinatorica 17 (1997), 459–467.
J. Fox, J. Pach, A. Sheffer, A. Suk and J. Zahl: A semi-algebraic version of Zarankiewicz’s problem, J. Eur. Math. Soc., to appear. Preprint arXiv:1407.5705, 2015.
L. Guth and N. Katz: On the Erdős distinct distances problem in the plane, Ann. Math. 181 (2015), 155–190.
P. Kővári, V. Sós and P. Turán: On a problem of Zarankiewicz, Colloq. Math. 3 (1954), 50–57.
K. F. Roth: On certain sets of integers, J. London Math. Soc. 28 (1953), 104–109.
G. Sárközy and S. Selkow: On edge colorings with at least q colors in every subset of p vertices, Electron. J. Combin. 8 (2001), no. 1.
A. Sheffer: Distinct distances: open problems and current bounds, preprint, arXiv:1406.1949, 2015.
A. Sheffer: Lower bounds for incidences with hypersurfaces, preprint, arXiv: 1511:03298, 2015.
E. Szemerédi: On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.
V. Vizing: On an estimate of the chromatic class of a p-graph, Diskret. Analiz. 3 (1964), 25–30.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by a Packard Fellowship, by NSF CAREER award DMS-1352121, and by an Alfred P. Sloan Fellowship.
Research partially supported by Swiss National Science Foundation grants 200020-165977 and 200021-162884.
Supported by NSF grant DMS-1500153.
