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Part of 1984 Miklós Schweitzer

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Miklós Schweitzer 1984- Problem 4

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9/4/2016
4. Let x1,x2,y1,y2,z1,z2x_1 , x_2 , y_1 , y_2 , z_1 , z_2 be transcendental numbers. Suppose that any 3 of them are algebraically independent, and among the 15 four-tuples on {x1,x2,y1,y2}\{x_1 , x_2 , y_1, y_2 \}, {x1,x2,z1,z2}\{ x_1 , x_2 , z_1 , z_2 \} and {y1,y2,z1,z2} \{y_1 , y_2 , z_1 , z_2 \} are algebraically dependent. Prove that there exists a transcendental number tt that depends algebraically on each of the pairs {x1,x2}\{ x_1 , x_2\} , {y1,y2}\{ y_1 , y_2 \}, and {z1,z2}\{ z_1 , z_2 \}. (A.37) [L. Lovász]
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