Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Abstract   Given a linear equation   L  , a set   A  ⊆  [  n  ]   is   L  -free if  A  does not contain any ‘non-trivial’ solutions to   L  .   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?     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,  ansr (ii) for a wide class of such equations   L  , thereby refining a special case of a result of Green [15].  also give various bounds on the number of maximal   L  -free subsets of [ n ] for three-variable homogeneous linear equations   L  . For this,  make use of container and removal lemmas of Green [15].
['Robert Hancock', 'Andrew Treglown']
I am writing a research paper on a idea Enumerating solution-free sets in the integers
[{"from": "human", "value": "I am writing a research paper on a idea Enumerating solution-free sets in the integers"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Abstract   Given a linear equation   L  , a set   A  \u2286  [  n  ]   is   L  -free if  A  does not contain any \u2018non-trivial\u2019 solutions to   L  .   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?     completely resolve (i) in the case when   L   is the equation   p  x  +  q  y  =  z   for fixed   p  ,  q  \u2208  N   where   p  \u2265  2  . Further, up to a multiplicative constant,  ansr (ii) for a wide class of such equations   L  , thereby refining a special case of a result of Green [15].  also give various bounds on the number of maximal   L  -free subsets of [ n ] for three-variable homogeneous linear equations   L  . For this,  make use of container and removal lemmas of Green [15]."}]