Decomposing sets without arithmetic progressions

Let A be a subset of \lbrace 1,2, \ldots ,8 \rbrace containing no arithmetic progressions. Then one of the following two must hold :

(1) |A \cap \lbrace 3,5 \rbrace | \leq 1 and |A \cap \lbrace 1,2,4,6,7,8 \rbrace | \leq 1.

(2) |A \cap \lbrace 4,6 \rbrace | \leq 1 and |A \cap \lbrace 1,2,3,5,7,8 \rbrace | \leq 1.

Let A be a subset of \lbrace 1,2, \ldots ,10 \rbrace containing no arithmetic progressions. Then one of |A \cap  \lbrace 1,2, \ldots ,7 \rbrace| and |A \cap  \lbrace 4,5, \ldots ,10 \rbrace| is at most $3$.

Let A be a subset of \lbrace 1,2, \ldots ,13 \rbrace containing no arithmetic progressions. Then one of the following two must hold :

(1) |A \cap \lbrace 1,2,3,4,6 \rbrace | \leq 3 and |A \cap \lbrace 5,7,8,9,10,11,12,13 \rbrace | \leq 4.

(2) |A \cap \lbrace 8,10,11,12,13 \rbrace | \leq 3 and |A \cap \lbrace 1,2,3,4,5,6,7,9 \rbrace | \leq 4.

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