#### Vol. 12, No. 6, 2019

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Sidon sets and 2-caps in $\mathbb{F}_3^n$

### Yixuan Huang, Michael Tait and Robert Won

Vol. 12 (2019), No. 6, 995–1003
##### Abstract

For each natural number $d$, we introduce the concept of a $d$-cap in ${\mathbb{F}}_{3}^{n}$. A set of points in ${\mathbb{F}}_{3}^{n}$ is called a $d$-cap if, for each $k=1,2,\dots ,d$, no $k+2$ of the points lie on a $k$-dimensional flat. This generalizes the notion of a cap in ${\mathbb{F}}_{3}^{n}$. We prove that the $2$-caps in ${\mathbb{F}}_{3}^{n}$ are exactly the Sidon sets in ${\mathbb{F}}_{3}^{n}$ and study the problem of determining the size of the largest $2$-cap in ${\mathbb{F}}_{3}^{n}$.

##### Keywords
Sidon sets, cap sets, caps, 2-caps
##### Mathematical Subject Classification 2010
Primary: 05B10, 05B25, 05B40, 51E15