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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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 .