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Abstract
We determine the maximum number of points in
ℝ d which form
exactly
t
distinct triangles, where we restrict ourselves to the case of
t
= 1 . We denote this
quantity by
F d ( t ) .
It is known from the work of Epstein et al. (Integers 18 (2018), art. id. A16 ) that
F 2 ( 1 )
= 4 . Here we show somewhat
surprisingly that
F 3 ( 1 )
= 4
and
F d ( 1 )
=
d
+ 1 ,
whenever
d
≥ 3 ,
and characterize the optimal point configurations. This is an extension of a variant of
the distinct distance problem put forward by Erdős and Fishburn (Discrete Math.
160 :1-3 (1996), 115–125 ).
Keywords
one-triangle problem, Erdős problem, optimal
configurations, finite point configurations
Mathematical Subject Classification 2010
Primary: 52C10
Secondary: 52C35
Milestones
Received: 12 February 2019
Revised: 29 September 2019
Accepted: 11 November 2019
Published: 4 February 2020
Communicated by Kenneth S. Berenhaut