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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. We determine the precise size of the largest Lfree subset of [n] for several general
classes of linear equations L of the form px+qy = rz for fixed p, q, r ∈ N where p ≥ q ≥ r. Further,
for all such linear equations L, we give an upper bound on the number of maximal Lfree subsets of
[n]. In the case when p = q ≥ 2 and r = 1 this bound is exact up to an error term in the exponent.
We make use of container and removal lemmas of Green [12] to prove this result. Our results also
extend to various linear equations with more than three variables.
Original language  English 

Pages (fromto)  1533 
Journal  Acta Arithmetica 
Volume  180 
Issue number  1 
Early online date  1 Aug 2017 
DOIs  
Publication status  Published  Sept 2017 
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Dive into the research topics of 'On solutionfree sets of integers II'. Together they form a unique fingerprint.Projects
 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