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On solution-free sets of integers.
- Source :
-
European Journal of Combinatorics . Dec2017, Vol. 66, p110-128. 19p. - Publication Year :
- 2017
-
Abstract
- Given a linear equation L , a set A ⊆ [ n ] is L -free if A does not contain any ‘non-trivial’ solutions to L . In this paper we consider the following three general questions: (i) What is the size of the largest L -free subset of [ n ] ? (ii) How many L -free subsets of [ n ] are there? (iii) How many maximal L -free subsets of [ n ] are there? We completely resolve (i) in the case when L is the equation p x + q y = z for fixed p , q ∈ N where p ≥ 2 . Further, up to a multiplicative constant, we answer (ii) for a wide class of such equations L , thereby refining a special case of a result of Green (2005). We also give various bounds on the number of maximal L -free subsets of [ n ] for three-variable homogeneous linear equations L . For this, we make use of container and removal lemmas of Green (2005). [ABSTRACT FROM AUTHOR]
- Subjects :
- *INTEGERS
*SET theory
*LINEAR equations
*MATHEMATICS
*COMBINATORICS
Subjects
Details
- Language :
- English
- ISSN :
- 01956698
- Volume :
- 66
- Database :
- Academic Search Index
- Journal :
- European Journal of Combinatorics
- Publication Type :
- Academic Journal
- Accession number :
- 124936527
- Full Text :
- https://doi.org/10.1016/j.ejc.2017.06.018