ON SIDON SETS IN A RANDOM SET OF VECTORS

Title & Authors
ON SIDON SETS IN A RANDOM SET OF VECTORS
Lee, Sang June;

Abstract
For positive integers d and n, let $\small{[n}$$\small{]}$$\small{^d}$ be the set of all vectors ($\small{a_1,a_2,{\cdots},a_d}$), where ai is an integer with $\small{0{\leq}a_i{\leq}n-1}$. A subset S of $\small{[n}$$\small{]}$$\small{^d}$ is called a Sidon set if all sums of two (not necessarily distinct) vectors in S are distinct. In this paper, we estimate two numbers related to the maximum size of Sidon sets in $\small{[n}$$\small{]}$$\small{^d}$. First, let $\small{\mathcal{Z}_{n,d}}$ be the number of all Sidon sets in $\small{[n}$$\small{]}$$\small{^d}$. We show that $\small{{\log}(\mathcal{Z}_{n,d})={\Theta}(n^{d/2})}$, where the constants of $\small{{\Theta}}$ depend only on d. Next, we estimate the maximum size of Sidon sets contained in a random set $\small{[n}$$\small{]}$$\small{^d_p}$, where $\small{[n}$$\small{]}$$\small{^d_p}$ denotes a random set obtained from $\small{[n}$$\small{]}$$\small{^d}$ by choosing each element independently with probability p.
Keywords
Sidon set;Sidon sequence;vector;
Language
English
Cited by
1.
The number of B3-sets of a given cardinality, Journal of Combinatorial Theory, Series A, 2016, 142, 44
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