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A Collection of Results of Simonyi's Conjecture

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Title: A Collection of Results of Simonyi's Conjecture
Author: Styner, Dustin
Department: Department of Mathematics and Statistics
Program: Mathematics and Statistics
Advisor: Pereira, RajeshMcNicholas, Paul
Abstract: Given two set systems $\mathscr{A}$ and $\mathscr{B}$ over an $n$-element set, we say that $(\mathscr{A,B})$ forms a recovering pair if the following conditions hold: \\ $ \forall A, A' \in \mathscr{A}$ and $ \forall B, B' \in \mathscr{B}$, $A \setminus B = A' \setminus B' \Rightarrow A=A'$ \\ $ \forall A, A' \in \mathscr{A}$ and $ \forall B, B' \in \mathscr {B}$, $B \setminus A = B' \setminus A' \Rightarrow B=B'$ \\ In 1989, G\'{a}bor Simonyi conjectured that if $(\mathscr{A,B})$ forms a recovering pair, then $|\mathscr{A}||\mathscr{B}|\leq 2^n$. This conjecture is the focus of this thesis. This thesis contains a collection of proofs of special cases that together form a complete proof that the conjecture holds for all values of $n$ up to 8. Many of these special cases also verify the conjecture for certain recovering pairs when $n>8$. We also present a result describing the nature of the set of numbers over which the conjecture in fact holds. Lastly, we present a new problem in graph theory, and discuss a few cases of this problem.
URI: http://hdl.handle.net/10214/4926
Date: 2012-12


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