Got a directed graph? I can give you at least one directed graph of size 3.

Theorem: There are n points. Each pair of distinct points is joined with an arrow. There is a cycle of length k for some k ∈ {3, 4, ...., n}. Then there must exist a cycle of length 3.

Proof: Suppose we have n points and all points have relations (arrows) with each other and there exists a cycle of size k. If there exists a cycle of length 3, the result is trivially true. Now suppose k > 3. Now what we do is, choose any (k ? 1) points or sides. There will be remaining one point and its adjacent two points. With these three points, we get a triangle, if it forms a cycle we stop. If it doesn’t, again from that (k ? 1) polygon we choose any (k ? 2) points or sides. There is one remaining point again and it’s two adjacent points, if this forms a cycle, we stop. If it doesn’t we again smaller or partitions the polygon.Like this (k ? 1, k ? 2, ......k ? r) such that k ? r = 4. For a quadrilateral, we will show the result.

In both the figure, the pink shaded position is the k = 3 cycled. Hence, it generalizes for any n and k.

(This proof is open for peer review :D)