Sayan Condition

Discrete Mathematics Level 2

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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