Sayan Condition

Determine the least positive integer n n for which the following condition holds: No matter how the elements of the set of the first nn positive integers, i.e. {1,2,n} \{1, 2, \ldots n \}, are colored in red or blue, there are (not necessarily distinct) integers x,y,z x, y, z, and w w in a set of the same color such that x+y+z=w x + y + z = w.

Details and assumptions

The phrase not necessarily distinct means that the integers can be repeated. For example, if 1,2,41, 2, 4 are all colored red, then we have 1+1+2=4 1 + 1 + 2 = 4 which would satisfy the condition.

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