Let be a subset of containing no arithmetic progressions. Then one of the following seven must hold :
(1) .
(2) .
(3) for any .
(4) for any .
(5) for any .
(6) for any .
(7) for any .
Let be a subset of containing no arithmetic progressions. Then one of the following seven must hold :
(1) .
(2) .
(3) for any .
(4) for any .
(5) for any .
(6) for any .
(7) for any .
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