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Abstract
Given a linear equation L, a set A ⊆ [n] is Lfree if A does not contain any ‘nontrivial’ solutions to L. In this paper we consider the following three general questions:
(i) What is the size of the largest Lfree subset of [n]?
(ii) How many Lfree subsets of [n] are there?
(iii) How many maximal Lfree subsets of [n] are there?
We completely resolve (i) in the case when L is the equation px + qy = 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 [15]. We also give various bounds on the number of maximal Lfree subsets of [n] for threevariable homogeneous linear equations L. For this, we make use of container and removal lemmas of Green [15].
(i) What is the size of the largest Lfree subset of [n]?
(ii) How many Lfree subsets of [n] are there?
(iii) How many maximal Lfree subsets of [n] are there?
We completely resolve (i) in the case when L is the equation px + qy = 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 [15]. We also give various bounds on the number of maximal Lfree subsets of [n] for threevariable homogeneous linear equations L. For this, we make use of container and removal lemmas of Green [15].
Original language  English 

Pages (fromto)  2130 
Journal  Electronic Notes in Discrete Mathematics 
Volume  56 
DOIs  
Publication status  Published  1 Dec 2016 
Keywords
 Container method
 independent sets
 solutionfree sets
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 1 Finished

EPSRC Fellowship: Dr Andrew Treglown  Independence in groups, graphs and the integers
Engineering & Physical Science Research Council
1/06/15 → 31/05/18
Project: Research Councils