Set systems with distinct sumsets

Q2 Mathematics

Electronic Notes in Discrete Mathematics Pub Date : 2018-07-01 DOI:10.1016/j.endm.2018.06.004

Javier Cilleruelo , Oriol Serra , Maximilian Wötzel

引用次数: 0

Abstract

A family $A$ of k-subsets of ${1, 2, \dots, N}$ is a Sidon system if the sumsets $A + A^{'}, A, A^{'} \in A$ are pairwise distinct. We show that the largest cardinality $F_{k} (N)$ of a Sidon system of k-subsets of $[N]$ satisfies $F_{k} (N) \leq (\begin{matrix} N - 1 \\ k - 1 \end{matrix}) + N - k$ and the asymptotic lower bound $F_{k} (N) = Ω_{k} (N^{k - 1})$ . More precise bounds on $F_{k} (N)$ are obtained for $k \leq 3$ . We also obtain the threshold probability for a random system to be Sidon for $k = 2$ and 3.

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集合具有不同集合的系统

如果集合A+A '，A,A '∈A是两两不同的，则由{1,2，…，N}的k个子集组成的族A是一个西顿系统。证明了由[N]的k个子集组成的Sidon系统的最大基数Fk(N)满足Fk(N)≤(N−1k−1)+N−k，且其渐近下界Fk(N)=Ωk(Nk−1)。当k≤3时，得到了Fk(N)上更精确的界。我们还得到了k=2和3时随机系统为Sidon的阈值概率。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics

CiteScore

1.30

自引率

0.00%

发文量

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