The maximum cardinality of a planar seven-distance set

Alon Amit

The maximum cardinality of a planar seven-distance set

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Contributed by Alon Amit ↗

Version 1 / Oct 26, 2026 / CC BY 4.0

Abstract

We give a computer-assisted proof that the maximum cardinality of a planar set determining at most seven distinct positive distances is sixteen. A diameter-graph argument reduces a hypothetical seventeen-point example to an eleven-point set with at most six distances, except for a thirteen-cycle obstruction. We exclude the cycle by an exact interval certificate. Using a published exhaustive count of sixty-two candidate distance patterns, we classify the eleven-point cores by exact Gröbner-basis calculations and prove that none can have the required seventeen- point extension. All extension calculations use rational or algebraic arithmetic. An explicit sixteen-point triangular-lattice set attains the bound.

Tools used

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References

P. Erdős and P. Fishburn, Maximum planar sets that determine $k$ distances, Discrete Mathematics 160 (1996), 115--125. doi:10.1016/0012-365X(95)00153-N (https://doi.org/10.1016/0012-365X(95)00153-N).DOI
X. Wei, A proof of Erdős--Fishburn's conjecture for $g(6)=13$ , Electronic Journal of Combinatorics 19 (4) (2012), P38. doi:10.37236/2917 (https://doi.org/10.37236/2917).DOI
F. Szöllősi and P. R. J. Östergå rd, Constructions of maximum few-distance sets in Euclidean spaces, Electronic Journal of Combinatorics 27 (1) (2020), P1.23. doi:10.37236/8565 (https://doi.org/10.37236/8565).DOI
W. T. Budianto, An Observation on 7-distance Set in Euclidean Plane , master's thesis, Department of Mathematics, National Central University, Taiwan, June 2022. https://ir.lib.ncu.edu.tw/handle/987654321/88394.link
L.-R. Bao and W.-H. Yu, Constructions of Large $m$ -Distance Sets on Triangular Lattice, arXiv:2509.00880v1 (2025). https://arxiv.org/abs/2509.00880.arXiv

Version history

v1Submitted by Alon AmitInitial depositCurrentOct 25, 2026