Raphael Walker | Searcharxiv

arXiv2022

How many cards should you lay out in a game of EvenQuads?: A detailed study of caps in AG(n,2)

We define a \textit{cap} in the affine geometry $AG(n,2)$ to be a subset in which any collection of 4 points is in general position. In this paper we classify, up to affine equivalence, all caps in $AG(n,2)$ of size $k \leq 9$. As a result, we obtain a complete characterization of caps in dimension $n \leq 6$, in particular complete and maximal caps. Since the \textit{EvenQuads} card deck is a model for $AG(6,2)$, as a consequence we determine the probability that an arbitrary $k$-card layout contains a quad.

math.CO↗

arXiv2021

A Small Maximal Sidon Set In $Z_2^n$

A Sidon set is a subset of an Abelian group with the property that each sum of two distinct elements is distinct. We construct a small maximal Sidon set of size $O((n \cdot 2^n)^{1/3})$ in the group $\mathbb{Z}_2^n$, generalizing a result of Ruzsa concerning maximal Sidon sets in the integers.

math.CO↗