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On the Number of Bh-Sets

Part of: Extremal combinatorics Combinatorics Sequences and sets

Published online by Cambridge University Press:  16 September 2015

DOMINGOS DELLAMONICA Jr ,
YOSHIHARU KOHAYAKAWA ,
SANG JUNE LEE ,
VOJTĚCH RÖDL  and
WOJCIECH SAMOTIJ
DOMINGOS DELLAMONICA Jr
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: domingos.junior@gmail.com, rodl@mathcs.emory.edu)
YOSHIHARU KOHAYAKAWA
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: domingos.junior@gmail.com, rodl@mathcs.emory.edu) Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, 05508-090 São Paulo, Brazil (e-mail: yoshi@ime.usp.br)
SANG JUNE LEE
Affiliation:
Department of Mathematics, Duksung Women's University, Seoul 132-714, South Korea (e-mail: sanglee242@duksung.ac.kr, sjlee242@gmail.com)
VOJTĚCH RÖDL
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30322, USA (e-mail: domingos.junior@gmail.com, rodl@mathcs.emory.edu)
WOJCIECH SAMOTIJ
Affiliation:
School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel and Trinity College, Cambridge CB2 1TQ, UK (e-mail: samotij@post.tau.ac.il)
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Abstract

A set A of positive integers is a Bh-set if all the sums of the form a1 + . . . + ah, with a1,. . .,ah ∈ A 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.

MSC classification

Primary: 11B75: Other combinatorial number theory 05A16: Asymptotic enumeration
Secondary: 05D40: Probabilistic methods 11B83: Special sequences and polynomials

Information

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 25 , Issue 1: Oberwolfach Special Issue Part 2 , January 2016 , pp. 108 - 129
Copyright
Copyright © Cambridge University Press 2015 

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