#### 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 ${ℝ}^{d}$ which form exactly $t$ distinct triangles, where we restrict ourselves to the case of $t=1$. We denote this quantity by ${F}_{d}\left(t\right)$. It is known from the work of Epstein et al. (Integers 18 (2018), art. id. A16) that ${F}_{2}\left(1\right)=4$. Here we show somewhat surprisingly that ${F}_{3}\left(1\right)=4$ and ${F}_{d}\left(1\right)=d+1$, whenever $d\ge 3$, 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
Primary: 52C10
Secondary: 52C35