# Gwen Spencer

Harvey Mudd College Mathematics 2005

Completed Thesis: | Combinatorial Consequences of relatives of the Lusternik-Schnirelmann-Borsuk Theorem |
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Appendix A: | A paper in progress on work described in my thesis |

Appendix B: | The Lusternik-Schnirelmann-Borsuk Theorem Implies the KKM Lemma |

Thesis Advisor: | Professor Francis Edward Su |

Second Reader: | Professor A. Castro |

E-Mail: | gspencer at hmc dot edu |

## Combinatorial Consequences of relatives of the Lusternik-Schnirelmann-Borsuk Theorem

Call a set of 2n + k elements Kneser-colored when its n-subsets are put into classes such that disjoint n-subsets are in different classes. Kneser showed that k + 2 classes are sufficient to Kneser-color the n-subsets of a 2n + k element set. There are several proofs that this same number is necessary which rely on fixed-point theorems related to the Lusternik-Schnirelmann- Borsuk (LSB) theorem. By employing generalizations of these theorems we expand the proofs mentioned to obtain proofs of an original result we call the Subcoloring theorem. The Subcoloring theorem asserts the existence of a partition of a Kneser-colored set that halves its classes in a special way. We demonstrate both a topological proof and a combinatorial proof of this main result. We present an original corollary that extends the Subcoloring theorem by providing bounds on the size of the pieces of the asserted partition. Throughout, we formulate our results both in combinatorial and graph theoretic terminology.

For perspective on the evolution of my thesis, here is my original thesis proposal: Original Thesis Proposal.

Here are the slides of a presentation I gave on this topic during the spring of 2005: Presentation Slides.

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## Future Plans

During 2005-2006 I will be travelling on a Watson Fellowship. Here is a link to the Watson Fellowship Homepage and a nice wirte up about my project:

In the fall of 2006 I will enroll at Cornell University in the Operations Research and Industrial Engineering Depertment where I will study math, optimization and statistics. For an explanation of what Operations Research is and informaiton about the department I'll be joining, see