Skip to main content

On the Number of Bh-Sets


A set A of positive integers is a Bh-set if all the sums of the form a1 + . . . + ah, with a1,. . .,ahA and a1 ⩽ . . . ⩽ ah, are distinct. We provide asymptotic bounds for the number of Bh-sets of a given cardinality contained in the interval [n] = {1,. . .,n}. As a consequence of our results, we address a problem of Cameron and Erdős (1990) in the context of Bh-sets. We also use these results to estimate the maximum size of a Bh-sets contained in a typical (random) subset of [n] with a given cardinality.

Hide All
[1]Alon, N., Balogh, J., Morris, R. and Samotij, W. (2014) Counting sum-free sets in Abelian groups. Israel J. Math. 199 309344.
[2]Bose, R. C. and Chowla, S. (1962/1963) Theorems in the additive theory of numbers. Comment. Math. Helv. 37 141147.
[3]Cameron, P. J. and Erdős, P. (1990) On the number of sets of integers with various properties. In Number Theory (Banff 1988), de Gruyter, pp. 61–79.
[4]Chen, S. (1994) On the size of finite Sidon sequences. Proc. Amer. Math. Soc. 121 353356.
[5]Chowla, S. (1944) Solution of a problem of Erdős and Turán in additive-number theory. Proc. Nat. Acad. Sci. India. Sect. A 14 12.
[6]Cilleruelo, J. (2001) New upper bounds for finite Bh sequences. Adv. Math. 159 117.
[7]Conlon, D. and Gowers, W. T. (2010) Combinatorial theorems in sparse random sets. Submitted.
[8]D'yachkov, A. G. and Rykov, V. V. (1984) Bs-sequences. Mat. Zametki 36 593601.
[9]Erdős, P. (1944) On a problem of Sidon in additive number theory and on some related problems: Addendum. J. London Math. Soc. 19 208.
[10]Erdős, P. and Turán, P. (1941) On a problem of Sidon in additive number theory, and on some related problems. J. London Math. Soc. 16 212215.
[11]Green, B. (2001) The number of squares and Bh[g] sets. Acta Arith. 100 365390.
[12]Halberstam, H. and Roth, K. F. (1983) Sequences, second edition, Springer.
[13]Hardy, G. H., Littlewood, J. E. and Polya, G. (1934) Inequalities, Cambridge University Press.
[14]Janson, S., Łuczak, T. and Ruciński, A. (2000) Random Graphs, Wiley-Interscience.
[15]Jia, X. D. (1993) On finite Sidon sequences. J. Number Theory 44 8492.
[16]Kleitman, D. J. and Wilson, D. B. (1996) On the number of graphs which lack small cycles. Unpublished Manuscript.
[17]Kleitman, D. J. and Winston, K. J. (1982) On the number of graphs without 4-cycles. Discrete Math. 41 167172.
[18]Kohayakawa, Y., Lee, S. J., Rödl, V. and Samotij, W. (2015) The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers. Random Struct. Alg. 46 125.
[19]Kohayakawa, Y., Lee, S. and Rödl, V. (2011) The maximum size of a Sidon set contained in a sparse random set of integers. In Proc. Twenty-Second Annual ACM–SIAM Symposium on Discrete Algorithms (Philadelphia, PA), SIAM, pp. 159–171.
[20]Kohayakawa, Y., Łuczak, T. and Rödl, V. (1996) Arithmetic progressions of length three in subsets of a random set. Acta Arith. 75 133163.
[21]Kolountzakis, M. N. (1996) The density of Bh[g] sequences and the minimum of dense cosine sums. J. Number Theory 56 411.
[22]Krückeberg, F. (1961) B 2-Folgen und verwandte Zahlenfolgen. J. Reine Angew. Math. 206 5360.
[23]Lee, S. J. On Sidon sets in a random set of vectors. J. Korean Math. Soc., to appear.
[24]Lindström, B. (1969) A remark on B 4-sequences. J. Combin. Theory 7 276277.
[25]O'Bryant, K. (2004) A complete annotated bibliography of work related to Sidon sequences. Electron. J. Combin. Dynamic Surveys 11 (electronic).
[26]Roth, K. F. (1953) On certain sets of integers. J. London Math. Soc. 28 104109.
[27]Saxton, D. and Thomason, A. (2015) Hypergraph containers, Invent. Math., 201 925992
[28]Schacht, M. Extremal results for random discrete structures. Submitted.
[29]Shparlinskii, I. E. (1986) On Bs-sequences. In Combinatorial Analysis, no. 7 (Russian), Moscow State University, pp. 4245.
[30]Singer, J. (1938) A theorem in finite projective geometry and some applications to number theory. Trans. Amer. Math. Soc. 43 377385.
[31]Szemerédi, E. (1975) On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 199245.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed