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Sperner's Lemma


Dissect a triangle into smaller triangles, such that all have full edge contact with their neighbors. Label the corners 1, 2, and 3. Label all vertices with 1, 2, or 3, with a restriction that the vertices of the side opposite a number lack that number. Thus, the side opposite 1 contains no vertices labelled 1.

Then Sperner's lemma states that any such labelling must contain an odd number of triangles with vertices labelled 1, 2, 3.

Sperner's Lemma is equivalent to the Brouwer fixed point theorem.


See also

Brouwer Fixed Point Theorem, Sperner's Theorem

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References

Harvey Mudd College Mathematics Department. "Mudd Math Fun Facts: Sperner's Lemma." http://www.math.hmc.edu/funfacts/ffiles/20001.4.shtml.Su, F. E. "Rental Harmony: Sperner's Lemma in Fair Division." Amer. Math. Monthly 106, 930-942, 1999.

Referenced on Wolfram|Alpha

Sperner's Lemma

Cite this as:

Weisstein, Eric W. "Sperner's Lemma." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SpernersLemma.html

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