The Extremal Principle in Graphs
Take the longest path, the vertex of largest degree.
Pick the extreme object (the longest edge, the smallest counterexample, the point closest to a line) and derive a contradiction or a forced structure. Because the extreme exists (finiteness or well-ordering guarantees it), the argument is airtight.
In combinatorics it drives most 'show some configuration must occur' proofs: take a vertex of maximum degree, a longest path, a triangle of least area.
Zeitz calls this one of the handful of genuinely universal problem-solving moves, and it pairs naturally with infinite descent.
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- Infinite Descent & Vieta JumpingNo smallest counterexample can survive.
- The Incircle ConfigurationTangent lengths turn sides into s − a.
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