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.

 

 

 

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