Vol. 13, No. 1, 2020

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Characterizing optimal point sets determining one distinct triangle

Hazel N. Brenner, James S. Depret-Guillaume, Eyvindur A. Palsson and Robert W. Stuckey

Vol. 13 (2020), No. 1, 91–98
Abstract

We determine the maximum number of points in d which form exactly t distinct triangles, where we restrict ourselves to the case of t = 1. We denote this quantity by Fd(t). It is known from the work of Epstein et al. (Integers 18 (2018), art. id. A16) that F2(1) = 4. Here we show somewhat surprisingly that F3(1) = 4 and Fd(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
Authors
Hazel N. Brenner
Department of Mathematics
Virginia Tech
Blacksburg, VA
United States
James S. Depret-Guillaume
Department of Mathematics
Virginia Tech
Blacksburg, VA
United States
Eyvindur A. Palsson
Department of Mathematics
Virginia Tech
Blacksburg, VA
United States
Robert W. Stuckey
Department of Mathematics
Virginia Tech
Blacksburg, VA
United States
Department of Mathematical Sciences
Kent State University
Kent, OH
United States