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The focus of this dissertation is on two problems in extremal set theory, which is a branch of extremal combinatorics. The general problem in extremal set theory is to start with all collections of subsets of an underlying ground set, apply restrictions, and then ask how large or small some property can be under those restrictions. We give a brief introduction to extremal combinatorics and consider two open questions.
One open question we consider is an extremal problem under “dimension constraints.” We give a brief account of the history of this subject and we consider the open problem of determining the maximum number of Hamming weight w vectors in a k-dimensional subspace of Fn2 . We determine this number for particular choices of n, k, and w, and provide a conjecture for the complete solution when w is odd. This problem is related to coding theory (the study of efficient transmission of data over noisy channels).
One tool used to study this problem is a linear map that decreases the weight of nonzero vectors by a constant. We characterize such maps. Using the tools we develop, we give a new elementary proof of the MacWilliams Extension Theorem (which characterizes weight-preserving linear maps).
The other open problem explored in this dissertation is related to a classical object known as a t-intersecting family, a set system where the size of the intersection of any two family members is at least t. The basic problem is to maximize the size of such a family. We give a history of the relevant theorems (with proofs, where appropriate).
A next question is how few pairs with intersection size less than t are possible in a (large) set system. Bollobás and Leader gave a new proof of a well-known partial solution to the t = 1 case by extending set systems to what they call fractional set systems. Although that paper claims the result for t > 1 in fact their generalization is false. In this dissertation we give give several counterexamples, as well as a fast algorithm to determine the minimizing fractional set systems when t > 1.
Advisor: Jamie Radcliffe