Sayan Condition

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

Details and assumptions

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

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