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 ${\log}(\mathcal{Z}_{n,d}) Keywords Sidon set;Sidon sequence;vector; Language English Cited by References 1. S. Chowla, Solution of a problem of Erdos and Turan in additive-number theory, Proc. Nat. Acad. Sci. India. Sect. A. 14 (1944), 1-2. 2. J. Cilleruelo, Sidon sets in${\mathbb{N}}^d$, J. Combin. Theory Ser. A 117 (2010), no. 7, 857-871. 3. P. Erdos, On a problem of Sidon in additive number theory and on some related problems. Addendum, J. Lond. Math. Soc. 19 (1944), 208. 4. P. Erdos and P. Turan, On a problem of Sidon in additive number theory, and on some related problems, J. Lond. Math. Soc. 16 (1941), 212-215. 5. H. Halberstam and K. F. Roth, Sequences, Second ed., Springer-Verlag, New York, 1983. 6. Y. Kohayakawa, S. Lee, and V. Rodl, The maximum size of a Sidon set contained in a sparse random set of integers, Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 159-171, SIAM, Philadelphia, PA, 2011. 7. Y. Kohayakawa, S. J. Lee, V. Rodl, and W. Samotij, The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers, Random Structures Algorithms 46 (2015), no. 1, 1-25. 8. B. Lindstrom, An inequality for$B_2$-sequences, J. Combin. Theory 6 (1969), 211-212. 9. B. Lindstrom, On$B_2\$-sequences of vectors, J. Number Theory 4 (1972), 261-265.

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