2016/07/28 by Robert Hancock, Hancock, Robert, Andrew Treglown +1
Mathematics · #05C69 #11B75 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05C69 #msc:11B75
paper · pdf · doi:10.48550/arxiv.1607.08399
20 pages, final version. To appear in European Journal of Combinatorics
arxiv created 2016/10/19 · arxiv updated 2016/10/20
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 px+qy=z for fixed p,q∈ \mathbb 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. 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.