## Sets without arithmetic progressions in [1..7]

Let $A$ be a subset of $\lbrace 1,2, \ldots ,7 \rbrace$ containing no arithmetic progressions. Then one of the following seven must hold :

(1) $A \cap \lbrace 1,4,7 \rbrace=\emptyset$.

(2) $A \cap \lbrace 3,4,5 \rbrace=\emptyset$.

(3) $|A \cap T | \leq 1$ for any $T\in \lbrace \lbrace 1,2,3 \rbrace, \lbrace 1,3,5 \rbrace\rbrace$.

(4)  $|A \cap T | \leq 1$ for any $T\in \lbrace \lbrace 3,5,7 \rbrace, \lbrace 5,6,7 \rbrace\rbrace$.

(5) $|A \cap T | \leq 1$ for any $T\in \lbrace \lbrace 1,3,5 \rbrace, \lbrace 1,4,7 \rbrace , \lbrace 3,4,5 \rbrace, \lbrace 4,5,6 \rbrace \rbrace$.

(6) $|A \cap T | \leq 1$ for any $T\in \lbrace \lbrace 1,4,7 \rbrace, \lbrace 3,5,7 \rbrace , \lbrace 2,3,4 \rbrace, \lbrace 3,4,5 \rbrace \rbrace$.

(7) $|A \cap T | \leq 1$ for any $T\in \lbrace \lbrace 2,3,4 \rbrace, \lbrace 3,4,5 \rbrace , \lbrace 4,5,6 \rbrace, \lbrace 2,4,6 \rbrace \rbrace$.